I've spent most of last year in the area, am in Colombia now, but here is some input.
I will be arriving at the Puebla airport (PBC) at approximately 8:00pm on a Saturday night. Is there reliable transporation available which would take me from the airport to the IMSS facilities on La Malinche?
Yes, but I would recommend to spend that Saturday night in Apizaco. It will save you unnecessary costs.
You cannot get that late to IMSS by bus. The cheapest would be to take the bus to Apizaco and get a taxi from there. But you'll get there late, better stay in Apizaco.
I've never flown to Puebla, but I think you will need to take a taxi to the CAPU bus terminal. From there you can still catch a bus with the company Atah to Apizaco, which is the closest town to Malinche to stay at. I don't know your budget, but if you are on your own, the IMSS is quite expensive, unless you are camping. There are cheap places to stayin Apizaco (like the one in front of the bus terminal). Then take the first bus to IMSS, if I remember well around 07am, a few blocks from the bus terminal.
Besides, I would travel to Tlaxcala afterwards, it's nice to walk around for an afternoon. I've climbed Malinche three times as a day hike from Apizaco. Don't know where you flying from and how sensitive to altitude you are though, so you might want to acclimatize. Then, yes: land in Puebla, taxi to the bus terminal, bus to Tlaxcala, spend the next morning walking around Tlaxcala, get to IMSS in the afternoon, climb Malinche the next day.
My last summit of Malinche was end of January, you needed crampons and ice axe then. You still might now.
If any more questions don't hesitate to contact me.