Learn Portuguese (Portugal): Eu quero outro café.

Yes, absolutely! mais um café literally translates to "more one coffee" (one more coffee). In fact, using mais um is the most natural and common way to ask for another round of drinks in Portugal. outro café is also completely correct, though it can sometimes imply you want a different coffee rather than just a second one.

Why isn't there a word for "a" or "an" before outro?

In English, "another" is a combination of "an" and "other". In Portuguese, the single word outro means "another" all by itself. You do not need to say um outro café, just outro café is enough.

Would outro change if I wanted another water instead of coffee?

Yes. Because outro is describing the drink, it must match the gender of that noun. café is a masculine word, so we use the masculine outro. If you wanted another water (água, which is feminine), you would need to say outra água.

Is it required to keep Eu (I) at the beginning of the sentence?