In my country Fajr time starts at 5.30 and I have to wake up at 6.30 to go to work. By waking up at 6.30 or 6.00 and then praying Fajr prayer, do I still get the same rewards as praying at 5.30 ?
As the comment by "Medi1Saif" mentions, we are unaware of the rewards of a deed.
That is to say, if there were a function
f: S -> R such that
S was the set of states of a person, and
R was the set of rewards attained, we currently do not know many facts about
However, it is possible to deduce the following:
Given two states
s_1,s_2\in S, if the only difference between
s_1involved a prayer done at the right time and
s_2involved that prayer done late, then
f(s_1) > f(s_2).
Proof: Suppose the contrary, that is, doing that prayer late gave you more or equal reward, then 17:78 would encourage one to do an action which lessens one's reward or does not modify it, which is a contradiction given the nature of the Coran (2:2) as a form of guidance. This completes the proof.
Therefore, everything else being equal (this is very important!), one does not get the same rewards when doing a prayer late, due to Theorem 1.