I'm doing the car free Friday commute from Mt Lebanon to downtown every Friday at 7:30. I'm going down the length of West Liberty so if you want to meet up somewhere along the route, let me know. If you are interested and Fridays don't work for you we can work out another day. You can email me with the private message link. That'll get you downtown at least. I cross over the Smithfield St bridge, but if you are heading down toward the Cardello building a neighbor used to prefer going up/down Greenleaf St at the west end of Mt Washington and over the West End bridge. I've never tried that myself. I'd recommend you don't take the Warrington/Arlington route.
As far as being in shape, I'd say if you can run 2 miles without stopping you are good to go over Mt Washington. If you can run one mile but not two you are probably fine if you have triple chain ring. It may take take you longer than you want at first, but you'll get over.
As others have said, you can hop on the T. There is a bike rack at South Hills junction, but you have to climb at least half the hill to get to it. I've been warned to make sure you have a good lock and lock your bike well but have never had any problems myself. It seems like there ought to be a way to lock up in the lot at the Palm Garden stop but I've never explored it. The 41E bus also runs in that area and seems to generally have racks on it although I don't think it is an official rack and roll route. If you want to avoid the hill completely you can get on the T at Dawn (up the cobbles off of W Liberty) but you have to lock up to the staircase railing. Not sure how safe it is there, but the neither the stop nor the road get much traffic.