I am aware that you must have control of a planet an entire game round before you can build on it.

I am similarly aware that the FAQ requires that there be one occupied planet in the system for each Saars base you wish to build.

My question is this would it be possible to build a spacedock in a system on the same turn you take control of it, if you successfully invade one of that systems planets?

No, from the Shattered Empire rulebook:

"Saar Space Docks may only be built in a system containing a planet that you have controlled for the entire game round."