If it would have been a general rule, like "always wash several cups, five or seven", then imperfective could be right, but here it sounds like an one-time order, so I guess perfective is more on it's place.
So, 'Тебе надо помыть некоторые чашки' would mean that from among the cups you had to wash certain ones? If the meaning is that you have to wash a small number, but no particular ones, the word 'несколько' is needed (?).
It is the Genitive and Prepositional form (also, the Accusative for animates). Несколько does not have the singular/plural distinction, though the case endings are the same as those of soft-stem adjectives in the plural (e.g., синих, синим, синими).