When was Ned fostered out to Jon Arryn? Would he have been around for Benjen’s birth? And do we know anything about what Rickard had planned for Benjen (like a fostering) before RR?

Ned was fostered when he was eight, and Benjen was probably born when Ned was six, so he would have been there for that. 

Given that Benjen was fourteen at the Tourney of Harrenhal and was still part of the Stark household, I’m guessing that he wasn’t fostered.