Is one allowed on Shabbos to dry the inside of a wet cup with a paper napkin, or would this be prohibited because of sechitah?
There are two melachoth that could be violated during an act of squeezing - dosh/mefarek (extracting) or melabein (cleaning). With regard to the former, your point is to absorb the moisture into the napkin, not to release it. If some was unintentionally extracted as well, that would be considered a davar she'eino mithkavein (unintentional) and therefore also permitted (even according to those who hold one would violate dosh/mefarek for desirable re-extraction of an artificially absorbed substance). With regard to melabein, the napkin typically is not desirably cleaner as a result of the drying of the cup. See, e.g., Shulchan Arukh OC 302:12, Mishna Berura 302:59, and Shemirath Shabbath Kehilchatha 12:19,21, all cited here.
If one did indeed intend/desire to clean a dirty cloth in the process, that would be forbidden. Similarly, if there is a likelihood that one would subsequently wring out and clean the drying cloth, there may be a rabbinic prohibition on using it. See e.g. Shulchan Arukh OC 302:11; Mishna Berura 302:46 and 302:57; and Shemirath Shabbath Kehilchatha 14:29, all cited here.