Any of the viscose "towels" or "wipes" will work, I have the Trader Joe's ones myself, and they not only make cleaning up a breeze, but make for excellent face cloths too:
Apartment Therapy Re-Nest | Feel The Love: Trader Joe's Super Amazing Kitchen Cloth Most Popular Posts
They're located in the store next to the kitchen stuff, dish soap, etc.
They make for good hand DRYING towels too, although they are small. Because they dry so quickly after using them, you never have to dry your hands with a wet towel.
They come in a random color pairing.
Similar items are sold at the drug store and other grocery stores.
You've probably seen this type too before:
Ecotech Envirolite Plus Cloth 50.0cm x 36.0cm Blue | EcoTech | Kitchen Hygiene
I have some in green, I think.
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There's always paper coasters.
307 White Coaster, Linen-Like, scalloped edge - 3 3/8 Inches ( HOFFMASTER876083 ), Hoffmaster 307 White Coaster, Linen-Like, scalloped edge
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I'd suggest both. Keep a couple of the viscose "towels" in the package, a couple zip-lock baggies (empty) and several paper coasters in another zip-lock baggie.
For the usual sweaty cup, the coaster will do the trick. Toss the used coaster(s) into the empty zip-lock for recycling/disposal (as there isn't always a trash receptacle or recycling bin nearby).
And in case of accidents, you whip out the larger absorbent towels, and let them do their thing to absorb, and wring them out.
You might also carry moistened hand wipes that can also be used for cleaning a bit of sticky from the cup holder, as well as cleaning your hands from wringing out the reusable viscose towel.
All the dampish things can go into the zip lock baggie until you get home.