I have learned that "bush" and "rush" are pronounced differently. Is there any rule to decide between these two pronunciations for similar words?
English does not have a phonetic alphabet. Instead of thinking that the spellings reflect pronunciations, you need to see that words have both spellings and pronunciations, and that they are not always consistent with each other.
One reason for this is that English spelling was mostly standardized during the Middle English period of the language. Since that time, the pronunciations have changed while the spellings have stayed the same.
Another reason is that the Latin alphabet is not ideal for the sounds of English. There are 26 letters, but there are 40+ sounds - 5 or 6 vowel letters but 16-21 vowel sounds, depending on dialect; and 20 or 21 consonant letters but 24 consonant sounds. Since English does not use diacritics, we use digraphs where possible. But it's a mess.
For the most part, English speakers learn the pronunciation first, and then memorize the spelling. This is the best idea for learners too.
So, even though bush and rush are spelled similarly, their vowels are different. For most English dialects, bush has the same vowel as look, and rush has the same vowel as up.
I can think of at least one -ush word where both pronunciations occur.
mush (meaning sloppy [food, etc.]) may be pronounced /mʌʃ/ (as rush) or /mʊʃ/ (as bush), though I suspect the first version (which is less common overall) is more likely to be encountered in Britain.
For other meanings (including the somewhat dated British slang face, mouth), it's always /mʌʃ/ (=push).
Regional accents also come into play. In the specific case of /ʌ/ and /ʊ/, speakers from the North of England use the second form in all words where Southerners use the former. That's why you normally hear blʌdy northerners (my SE form) bemoaning blʊdy sʊtherners (their NE form), and not vice-versa.
In short, this is just another illustration of the (incredibly awkward, I know) fact that there's no short definitive list of rules for determining pronunciation. Even native speakers occasionally make mistakes with words which they've only encountered in written rather than spoken form.