Check with your clinic, if they have transportaion doing pick-ups at the airport, you may just want to get back to the airport the following morning and use their transporation. Keeps it simple.
A cab to the border might cost you $20 or so. Otherwise, san diego has a light rail system I think they call the blueline trolley. Get on the train for San Ysidro/Tijuana and take it to the border station.
Walk across the border and catch a taxi on the otherside. Have the clinic name and address on a piece of paper to give the driver and have $5.00US ready. Taxi's in TJ travel to everywhere seem to cost $5. If you hand him a $10, the trip will cost $10. You won't get change. lol
I was told that doing it your self would take about two hours to get to the clinic. The clinic transport would be around a half hour from pick-up.
The border crossing going into Mx will be quick, nobody cares about people going into Mexico. Coming back will take much longer.
good luck