Drying the inside of a cup on Shabbos:
It is forbidden to dry a wet cup with a cloth [or tissue or napkin] on Shabbos. This applies whether the cup is wet due to water, wine or other beverages. This prohibition applies even against using a cloth which one is not particular against getting wet [such as a towel which is designated for drying]. [This prohibition applies by all cups, whether wide or narrow; whether a Kiddush cup, coffee cup or shot glass. It also applies irrelevant of how wet the cup is. However some Poskim learn it only applies to narrow cups, such as a shot glass. Likewise, some Poskim rule one may be lenient to dry the cup if one shakes the water out of the cup beforehand and it hence only contains a few drops. Some Poskim even allow drying with a cloth that is not designated for drying. Practically, those that are lenient to dry wide cups [such as a Kiddush cup and the like] should shake the water out beforehand.]
May one dry his Kiddush cup after rinsing it prior to Kiddush?
Based on the above, one is not to dry his Kiddush cup after rinsing it. Nevertheless, those which are lenient to shake out the water from Kiddush cup and then dry it have upon whom to rely, as stated above.
 302/12; Michaber 302/12; Tashbatz 28; Mahram 511; Rabbeinu Yerucham 12/13
 See Az Nidbaru 7/9; However according to the Igros Moshe 2/70 perhaps this would be allowed
 The reason: As since the cup is narrow, it is not possible to avoid the cloth from squeezing out some of the water which it absorbed while drying it, and it is thus Rabbinically forbidden [due to the squeezing/Mifarek prohibition] even by liquids that do not contain a laundering prohibition. [Admur ibid; M”A 302/27 in his second answer; M”B 302/60]
Other opinions: Some Poskim rule the reason for this prohibition is due to the laundering prohibition, and hence it is only prohibited by water or white wine, and only when using a cloth that is not designated for drying. [1st answer in M”A ibid; M”B 320/58] Some Poskim rule the custom is to be lenient to allow drying cups. [Ridbaz 1/213, brought in Birkeiy Yosef 302/89; M”B 302/59; See Biur Halacha “Mishum” that this applies even by narrow cups]
 Meaning that the prohibition applies even if the cup has a type of liquid to which the laundering prohibition is inapplicable. [ibid]
 Implication of Admur ibid and Michaber and M”A ibid which do not differentiate between the size of the cup; See Minchas Shabbos 80/129 and Misgeres Hashulchan 80/74 that negate limiting the prohibition to cups with narrow tops being the Poskim above did not differentiate in this; See Az Nidbaru 3/20 that tea and coffee cups should also be considered narrow cups although the custom is to be lenient; See Toras Hamelachos 13 Melaben
The reason: Seemingly the reason for this is because all cups are considered narrow enough to cause squeezing when drying its ends at the bottom. However a bowl or pot is wide enough to be dried without causing any squeezing.
 Implication of Admur ibid and Michaber ibid which do not differentiate between the wetness of the cup; Biur Halacha “Mishum” only allows to be lenient in a time of need if one shakes out the water
 SSH”K 12/22 footnote 61 [new]; See Kitzur SHU”A 80/34 “or other vessels that have a narrow opening”
 Or Letziyon 2/24-7; Az Nidbaru ibid that those that are lenient to dry coffee cups and the like should first shake out the water; Biur Halacha ibid is lenient to allow drying even a narrow cup in a time of need if one first shakes out the water.
 See M”A 302/27; M”B 302/59 which implies according to the Radbaz one may use even a non-designated cloth as there will not be inevitable squeezing, and the laundering prohibition does not apply when drying an item. Admur records a dispute in this matter in 302/21 and 319/23 and concludes that a G-d fearing Jew is to be stringent.
 Az Nidbaru ibid
 The Rebbes custom: Rabbi Groner related to me that the Rebbe dried his Kiddush cup by the weekly farbrengen only after the cup had been rinsed and already dried. The reason the Rebbe preceded to dry the already dried cup is based on that the Rebbe desired to personally clean his cup, and one can use a napkin in place of water. [See Kaf Hachaim 183/5]
 See M”B 302/59
 See Admur 613/16