"Eher trinke ich Milch als Orangensaft."
I agree with gewisse. As a native speaker, "I rather drink" sounds broken to my ears. Check out http://en.wiktionary.org/wiki/rather - the examples all use "I'd rather".
However, I'm mostly going by what I've heard in everyday speech. If anyone could provide a grammatical reason why this is incorrect, I would appreciate it.
In speech, it's probably totally fine to omit the "would" in "I would rather drink...", but I wouldn't do it in writing.
The only circumstance that "I rather..." is used in (that I can think of) is in a sentence like "I rather like that shade of blue", where "rather" means something more like "quite".
One might argue that "I rather drink milk than orange juice." = "Eher trinke ich Milch als Orangensaft." is different from "I would rather drink milk than orange juice." = "Ich würde eher Milch als Orangensaft trinken.
However, I think this would be overly strict, and I think one should equate all of the above.