Site A and B are both new, with few incoming links, and no 'do-follow' outgoing links.
Site A has PR1, and site B has PR0.
I am curious - Can anyone predict what might happen to the respective PRs if I add a link from A to B (but not reciprocate)?
It's not possible to say because we don't know enough about the PR1 web page or the PR0 web page (PageRank is per page, not per website). How many links are on the PR1 page? Is the link to you nofollow? Is the PR0 page close to a PR1? Or is it far away?
But even without knowing any of that, the answer is almost certainly, "nothing much". PR1 is a very low amount of PageRank. Even if the link to your site was the only link on that page it would be sending so little PageRank to you that all by itself it isn't going to do much for your page. If the page receiving the link is very close to being a PR1 then this link may put it over the edge.
What you should be focusing on is not PageRank but your pages' rankings in search results. Chasing PageRank is like chasing a ghost. It's a waste of time. Focus on building quality links to your pages. I'd rather have a link from a well-ranked on-topic page, than a poorly-ranked off-topic high PR page. On topic pages that are ranked well carry a lot more weight than an off-topic page with high PR. Google sees links as votes and the more authoritative the linking page (e.g. the higher ranked it is for that topic) the more weight its votes carries.