Can one wash for mayim ahachronim without a cup, i.e. just under the faucet lechatchilah?
It is permitted to wash one’s hands for mayim acharonim under the faucet.
Some, based on the Vilna Gaon, are particular to use a keli.
The Beis Yosef (181) and Mishnah Berurah (10, 21 and Shaar Ha-Tzion 31) explain that there is no need to be particular for the halachos of netilas yadayim for mayim acharonim.
See Shiltei Giborim, who does require a keli and other halachos even for mayim acharonim, but this is not ruled by the poskim.
The Mishnah Berurah does note that one should not use just a few drops of water, but use enough water to cover the first two vertabrae of the hand. However, the custom of using only a small amount of water is because our hands are generally clean, and the continued custom is mainly because of kabbalistic sources (this is why many do not do mayim acharonim today).
The Vilna Gaon writes that one should use a full revi’is of water, from a keli, but many write that there is no need for this, as noted above. See Ben Ish Chaim Shelach 9; Kaf Ha-Chaim 6; among others.