Washing Before Davening
There is a Halachic requirement to wash one’s hands before davening. This applies to all three prayers. There are two reasons for this cited in the Rishonim. One is that if your hands are dirty they must be washed before prayer. Additionally we suspect that one was distracted throughout the day and touched places that necessitate hand washing.
For Shacharis, if one was careful not to touch anything that would necessitate handwashing after his morning Netilas Yadayim then he need not wash again. The same applies to washing for Maariv if one davens Mincha and Maariv together.
If no water is available, and he definitely touched an area of the body that requires him to wash his hands, he must travel 18 (and sometimes up to 72) minutes to procure water, if by doing so he will still be able to Daven at the proper time.
In other situations he may wipe his hands on a cloth or something that cleanses if there is no water.
One should not wash in a bathroom, but if there is no choice he may do so, provided that he dry his hands outside the bathroom. (Igros Moshe EH 1:114)
When a Cup is Needed
The only times one must use a cup is in the morning and before eating bread. Rav Ben Zion Abba Shaul (Ohr litzion vol.2 1:5 in notes) infers from the Rambam that before prayer one should also use a vessel.
Although for all other items a cup is not needed, the Kaf Hachaim recommends using a cup for all the various required washings.