Theoretically, yes! Luna-1 should still be out there, slightly damaged by the solar wind and possible micrometeors but in otherwise totally intact condition. So in principle, it would be possible to send something out to grab Luna-1 and bring it back.
However this would be extremely difficult - even if you had the funding to do such a thing, finding a spacecraft of that size in such a massive area as Earth's solar orbit would be almost impossible. But, again, in principle, you could go to it. It's up there right now, probably looking pretty similar to how it did when it launched in 1959.
I think the issue will be working out where it is. Chances it will be found as an "asteroid" by the Large Synoptic Survey when it starts in the 2020's but will be impossible to link it back to Luna 2.
Poor luna-1. Lost forever to the void of space.