I have many Cubase 7 AI projects. If I get Cubase 8 Artist, will I be able to open that projects?
Yes, that will work just fine.
See this article for official Steinberg notes on compatibility (the image thumbnail doesn't seem to work, but it works if you click it). Here is the compatibility chart from the article:
In general the rule of thumb is that you can always open old projects created with a previous version in a newer version, but the other way around is more uncertain.