For some more magical or superstitious ways of fighting drunkenness off were amethyst stones. The very name "Amethyst" (Αμέθυστος) stands for "non-drunk" and is associated with god Dionysus. According to some traditions Rhea gave Dionysus the amethyst to cure him from his madness. This method seems to be very much loved by the ancient Romans as well for we see rings or even cups of amethyst being used to prevent drunkenness or hangover. (obviously needless to say that many people were up for a huge disappointment from that kind of cure hahahaha 😆) Also a papyrus discovered in Egypt in Greek text, dating back around 1900 years ago (from the team of papyri called Oxyrhynchus papyri) that seems to be containing a remedy for "drunken headache" that involves either a garland or a small portion of leaves from a plant named chamaedaphne (Dane racemosa) which was supposed to be worn, probably around the neck:

(papyrus's fragment and a sample of the plant in question)

In fact scents and aromas seemed to be preferred by the ancient Greeks to compact the fumes of alcohol that they believed caused hangover. Honestly the logical part of smelling plants or herbs seems to me to fight against the feeling of nausea than anything else.

Now for actual cures themselves there seems to be a number of sources from later years (4th-3rd century BC mostly) from various poets such as Alexis (Ἄλεξις) Eubulus (Εὔβουλος) or Sopater (Σώπατρος) but also of course on sources like Hippocrates or Aristotle or even Plutarch. Some cures suggested are:

Sleep it off! (yeah even the ancient Greeks knew that sleeping off your alcohol consumption seems step one). Already from Homer we hear examples like Polyphemus that lie down and sleep after the over-consumption of wine and later they are up again and about so obviously a good night's nap was step one here. Plutarch even mentions how someone needs to stay somewhere warm and in the morning he suggests that one should take a bath to come around and then try and eat something that doesn't agitate the stomach.

ἰῶνταί γε μὴν τὰς περὶ τὸ σῶμα τῶν μεθυσκομένων καὶ κραιπαλώντων κακώσεις εὐθὺς μὲν ὡς ἔοικε περιστολῇ καὶ κατακλίσει συνθάλποντες, μεθ᾿ ἡμέραν δὲ λουτρῷ καὶ ἀλείμματι καὶ σιτίοις, ὅσα μὴ ταράττοντα τὸν ὄγχον ἅμα πράως ἀνακαλεῖται τὸ θερμὸν ὑπὸ τοῦ οἴνου διεσπασμένον καὶ πεφυγαδευμένον ἐκ τοῦ σώματος.

To heal the body suffering from drinking and pains of hangover one can immediately go to sleep and cover themselves for warmth to feel better. And the next day one can take a bath and eat whatever foods will not unsettle their stomach but also restores the warmth that has lost by the wine (Translation by me)

Arguably this is a simple and honestly logical advise given how sleeping somewhere warm will help someone sweat it off and generally rest comfortably and a bath always helps the nausea and it is known that when someone is nauseating they need something to draw the liquids of the stomach so something soft to eat would do. Warm and calming foods also might as well help nausea. Swimming in the sea was also advised given how the iodine and the salt in the seawater is indeed known for its healing properties in antiquity and if one can go for a swim in the sea indeed helps a lot with nausea.

One famous cure for hangover seems to be the cabbage (either boiled or raw). Cabbage was believed to be an enemy to vine so both the Greeks and the Romans seemed to be suggesting it as a cure against drunkenness and hangover. Especially after a good nap:

ἐχθὲς ὑπέπινες, εἶτα νυνὶ κραιπαλᾷς. κατανύστασον· παύσῃ γάρ. εἶτά σοι δότω ῥάφανόν τις ἑφθήν. (Alexis fragment)

Yesterday you drank too much and now you are hangover. Go and get some sleep so it will subside. And then I shall give you some boiled cabbage. (Translation by me)

We need to say that the Greeks used the word ῥάφανόν for "cabbage" which seems to be closer to modern day kale than the usual cabbage (also associated with broccoli and cauliflower) we know or it was also doubling for the radish family. It seems to be a plant belonging to the family of Brassica cretica:

this plant in its wild form indeed has healing properties. I am not an archaeo-botanist to know in detail how much that plant was cultivated in Greece or if that plant is indeed mentioned in its wild form in the sources but yeah we can expect that the "cabbage" we see in ancient Greek texts is one of the ancestors of kale, broccoli and cauliflower.

Aristotle uses the same word as we do nowadays for the same plant "κράμβη" as you see from the image above. In fact he is mentioning other healing properties of the plant such as cure for diarrhea so he mentions the ability of the plant to "drink" the wet elements inside the stomach, including wine whilst he says the generally milder and sweeter flavor helps the unsettled stomach subside.

Hippocrates though has a different cure. He suggests a cup of undiluted wine for it:

Ἢν ἐκ κραιπάλης κεφαλὴν ἀλγέῃ, οἴνου ἀκρήτου κοτύλην πιεῖν· ἢν δὲ ἄλλως κεφαλὴν ἀλγέῃ, ἄρτον ὡς θερμότατον ἐξ οἴνου ἀκρήτου ἐσθίειν.

Whoever suffers from hangover headache should drink a cup of undiluted wine. For other type of headache one can eat bread warmed up with undiluted wine (Translation by me)

Ironically this cure is also suggested before that after a hangover one dosage of alcohol might bring you together. Most of the time I remember sweet and thicker wines were suggested for healing purposes (see for instance Pramnian wine especially in magical contexts like the potion of Circe). The sweetness or heavier taste of the undiluted wine seems to be given as a cure which does make sense but as Hippocrates says, it is just one cup (κοτύλη)

Something sweet (honey being suggested). So the sweetness and the soothing healing properties of honey seem to be also suggested for hangovers at least according to Sopater who notices how hangover men find it soothing to drink honey beverages the morning after their heavy drinking. This sweetness seems to be a key to the hangover recipes in general given how the previous methods could also be winking at some sweet thing like boiled cabbage juice to settle the sour-y feeling of the nausea or the stomach reflex. Moreover the sweet drinks can be drunk slowly and little by little so allows the body to heal while giving some calories and energy to the person who suffers from nausea

As for your last part saying my personal experience? Hahaha you wanna hear if I have had embarrassing hangover stories to tell? 😆 Well unfortunately I do hahahaha 😆I hate my hangovers because I get nausea that lasts for hours. I am mistress of my actions at all times but yes I throw up a lot and get dizzy not have headaches usually. So usually I follow the first advise hahaha 😆I usually try to get some sleep and get medication for nausea or eat something light or drink a cola or something to get my body to rest a bit somewhere warm and take a bath when I wake up to get better. Also following the advice of sweetness by drinking some cola helps to settle the nausea.

So I should say the few times that I was going through a hangover I am using a mix of modern and ancient remedies hahaha I do seem to be more settled to stay in bed for a few hours and I have noticed that a warm bath or sunbathing at the beach or swimming in the sea helps me a lot. Apart from modern nausea medication I do enjoy drinking something sweet (as I said usually cola is very helpful to me for stomach issues) and I like to eat some soft and warm food usually including wheat (for example we have a wheat and egg soup-like pasta that I like to sip slowly if I have stomachache)

So yes surprisingly these remedies are also common nowadays just added more modern media to them. To be fair I am not aware for many people eating boiled or raw kale the morning after their drinking and of course the common painkillers are more used these days but I know the aspect of sleeping your hangover off, eating something sweet or drinking it very very slowly is also very common as well as a nice warm bath. Swimming in the sea is advised for many people and for many reasons from ancient times till now because not only is the best form of exercise that is also helping pained muscles but also the minerals in sea water are known for being incredible healing factors. Now drinking a dosage of alcohol is something I have heard is common but I do not know many people who do it hahaha however sweet wine is being suggested many times and even nowadays is said that one moderate consumption of wine every day helps the heart function so a cup of red wine per day is said to be actually having healing properties for the body (I say, guilty to follow this parameter hahaha 😆)

However a modern healing method from drinking that we have is this rich soup created by the stomach of a cow alongside with other herbs and spicy flakes. We call it Patsas which is eaten typically after drinking too much:

Trust me it tastes 100 times better than what it sounds or smells (it has strong smell) haha also we see typically boiled meat such as boiled goat meat to be suggested for after a loud party. Again tried it myself is amazing:

which falls to the category of eating something warm that settles the stomach. And honestly judging by the fact that both us and our ancestors never wasted any part of the animal away I am willing to bet soups like that must have been used in antiquity too hehehe