A, no bullshit, hotshot archeologist comes face to face with a coldblooded tomb raider in a struggle to claim discovery of a lost Egyptian city of immeasurable wealth.

Is laying claim to the discovery really the conflict here? Sounds to me like the "tomb raider" wouldn't care if he was credited with the discovery -- only that he got his hands on the wealth. The archaeologist, on the other hand, would be desperate to keep the raider from destroying or stealing any priceless antiques (unless he's in it for the money as well).

I also think that you should choose between "hotshot" and "no bullshit", since otherwise it's a bit awkward at the start.

I would pick between 'no bullshit' and 'hotshot' to describe your main character, feels too wordy. The other thing I would think about is rephrasing your antagonist. The term 'tomb raider' has become synonymous with a single franchise, maybe grave robber? Here is my shot at it..

An ego driven archaeologist races against a coldblooded grave robber to lay claim to the treasures of a lost Egyptian city.