I think Mission Elevator was a budget release (which for 16-bit games means it sold for about £10, as opposed to the £25-£30 price range for all the full-price releases). It is a barefaced copy of Taito's Elevator Action coin-op. However, since the ST is considerably more advanced than the ancient coin-op, Mission Elevator has better graphics and (arguably) better sound. Unfortunately, it lacks the gameplay that makes the coin-op still enjoyable today - mainly, it's too difficult. Your character frisks about a hotel (?) full of elevators, searching objects and shooting guys in trench coats, with the goal of reaching the top of the building. That's what I believe, anyway - but I've never even reached the top of the first level in either the Amiga or ST version, because there are simply too many trench-coated villains shooting too rapidly.
Verdict: The graphics might be quite good, but the game is just too difficult. Don't bother with this one.