yes, it is a normal English expression, but not in the context of "I will update" It does not sound natural, because it means: without to much delay.
I will update without to much delay is not incorrect but does not sound natural, neither does , I will update sooner rather than later.
Maybe if you said, I would rather update sooner than later.
I would rather go shopping sooner than later.
English is a very complicated language and to make it more complicated there are many ways of speaking it depending on where you live in the U.S. Remember this is just my opinion and I respect others as well.