The Shulhan 'Arukh writes that cutlery, crockery and the like may be washed on Shabbath, provided that they will be needed for that Shabbath, such as when there are other S'euddoth (Shabbath meals) remaining. However, after Se'uddah Shelishith (the third Shabbat meal which is eaten after Minha), they may not be washed.
This implies that one may wash utensils after the Friday night meal for the Shabbath morning meal, or even for Se'uddah Shelishith, if they are needed. In addition, even if one requires only one cup, he may wash several since each of the ones washed would be fit to be used. However, if one has other clean utensils that could be used it is good to use the clean ones and not wash the dirty dishes on Shabbath because of the effort it requires.
The reason why we may not wash dishes and the like after Se'uddah Shelishith is because one does not normally eat after this, but if someone wishes to eat more, or guests are expected, then it is permitted.
(See Sh. 'A. 323: 6. Kaf Hahayim ibid, Oth 39-42)
