le can't be la hora because la hora is the subject. The sentence doesn't have enough information to determine who le refers to, but it could be usted, él, ella, or something else singular. Whenever you have multiple object pronouns, despite what many textbooks may say, regardless their function, the order is always se + 2nd person + 1st person + 3rd person (specifically, se, te, os, me, nos, le, les, lo, la, los, las — not all combinations are possible, obviously).
If you have the verb presentar, and you told me that a friend of ours introduced one of us to the other, the sentence actually end up 100% ambiguous: ¿no te acuerdas? Juan te me presentó. it could be you were introduced to me or I was introduced to you.
In any case, pasar has a number of reflexive uses including (from the DRAE)
pasar 55. prnl. Acabarse o dejar de ser.
Technically, this isn't reflexive in the sense that there's no action being performed back on the subject, but this usage is call pronominal, that is, it requires —always— a reflexive object, despite not being reflexive in meaning. In any case, we get pasarse meaning to end or conclude, and la hora as the subject, with le being done unspecified person.
So this sentence means, basically, time hasn't run out on him.