Very nice gun. I just bought (last weekend) an early model 35. It was serial number 401xx and was shipped in May of 1959. I think the 34 and 35's were serial numbered in the same sequence. You have a very neat gun. It should be on the improved I frame (coil spring). The model numbers weren't stamped until 1957 so yours should be a bit after that. Mine has an earlier serial number and wasn't shipped until 59...but you'll find ship dates all over the place sometimes with S&W.
I think yours had to have been made before 1960 because that's when they became 34-1's. That's also when they switched over and were made on the "J" frame. Your cylinder will be just a bit shorter than if it were a 34-1. I'm still learning about these little 22's so hopefully I'm not spouting incorrect info. The box looks correct and is an early pre bangor punta script. I'd say you have a very nice acquisition...and thanks for posting the pics.

Roger Epperson
SWCA #2175