Yes, "just over" modifies "a five-minute bike ride" just as "only" does.
I'm not so sure that we should leave "from the school" within the phrase. Actually, we can say:
Q: How far is the house from the school?
A: Only a five-minute bike ride.
A': Just over a five-minute bike ride.
The only difference I seem to find is that, even if it wouldn't be so usual or natural, "only" could be placed at the end:
A'': A five-minute bike ride only.
Instead, because of its typical usage with numbers (meaning "more than"), "(just) over" needs to appear before the number, as in A' or A''' below:
A''': A bike ride of just over five minutes.
Even if we can say "a bike ride of only five minutes," my impression is that, placed at the beginning of the phrase, "ony" will modify the whole phrase, so that if we want to make clear that it only restricts the length of the ride, we should say "of only five minutes."