I don't know if the rules say one way or another. I know 3 upgrades is the max but is there a limit to weapon upgrades or any other kind of upgrades?

I have the 2 player starter box that has Lhasa the cross guard lightsaber for Kylo Ren. I pull Kylo Ren's lightsaber from a booster. Can I put both weapon cards on him at same time?

You can put more that one weapon on a character. The upgrades are not limited by type. You just can't exceed three upgrades on a single character.

You also have to pay attention to whether the cards are unique. You can have both the Cross Guard and Kylo's Lightsaber at the same time because only Kylo's is unique, and they do not share the same title. But you could not play 2 Kylo's sabers at the same time.