Are there any turns in the stairs?

Have you folded down the backbox, put padding between the backbox and cabinet, strapped it down (or shrink wrapped it) and taken off the legs so the cab is standing up on end?

If it's just a straight staircase, strap the game to a hand truck or appliance hand truck that can handle at least 400lbs, and have at least 3-5 guys to drag it up.

Or, if there are any equipment rental places, a stair climbing hand truck, such as an escalera, will make the process much easier.

Or, yes, you can remove the backbox. Take careful photos of all the connectors...there are a few than can get mixed up. Disconnect everything feeding up from the cabinet. Undo the nut for the grounding wire. Unbolt the backbox from the hinges. Put some protection or cardboard between the hinges and the cab artwork. Then when everything is clear, remove the backbox. Then do everything in reverse order to put it back together later.

Keep in mind that there are a couple of connectors that can get mixed up. Also, if any of the connectors are hacked up, they may be in seperate pieces.