Is there a rule for us to know when to use “ύ" and when “β"?
I'd go with a general "no", unfortunately. There are some cases when you can tell, when you are able to guess from loanwords for example (1), or in compound words (2), but in general it's spelling that you have to memorise as is.
Yes, as kirakrakra says Υυ is a vowel and it sounds like the ee in "see" and gets an accent when it is on the syllable that is being stressed. For example: "σύνδεσμος" (connection).
You should note that in this word the υ is used in a diphthong...that is: 2 letters together that form a new sound " αυ it can be ev or **ef"".
You really need to see the Greek forum and all the information we give on the Greek alphabet and the language in general.
Go here: https://forum.duolingo.com/topic/936