wait,what? i have to pay to update a game i got for 13 bucks at a pawnshop?
Well if the game did not come with the key you're basically out of luck, just like most modern computer games. In general buying modern PC games used is not a good idea.
That said the 1.05 patch for the original Sins of a Solar Empire was the last to be available on the internet even without the CD key, and it was the first version with mod support. However, only some very old mods will be around for version 1.05, none of the cool mods you might be hearing about now are even for the original Sins, let alone version 1.05.
Also on the steam summer sale going on right now you could by Rebellion, the latest version of Sins with all the expansion packs (3 of them) for $13 bucks, so I'm afraid you just got a very bad deal. The original should have sold for less than $5 even if it was new with the CD key. Take it back if you can.