Learn Finnish: Aika on jo ohi. - The time is already over.

While asti and mennessä are used to set a deadline (meaning "until" or "by a certain time"), this sentence shows how to state that the deadline has been missed. The word ohi means "over" or "past," showing the time limit has expired.

Is ohi acting as a postposition in this sentence?

No, in this specific sentence ohi is simply an adverb meaning "over." However, it is very commonly used as a postposition governing the genitive case when talking about moving past a physical limit or location, such as kävelin talon ohi (I walked past the house).

Could I also say "Aika on jo loppu" to mean the same thing?

Yes, both are very common ways to say "time is up." Ohi focuses on the moment being "past" or "over," while loppu focuses on the allotted time being "finished" or "exhausted."