Breeze- I don't think you need to reserve yhe buggy. If you see one just ask the driver yo drop you at the gate or wherever you need to go. That's what I did. My son fell asleep and it was hard to carry him and the carryon. The person (driver) was happy to help us.
If you want to use this service, when you get out of the aircraft ask a gate agent to request a airport buggy, they will get you one. Get the best seat in flight before you get to the airport getting around the airport will be easy if you are able to relax on the flight. Which airline are you travelling with?
we going with emirates this time... i did travel earlier with this but not with kid.. have to see how it goes..
I flew with Qatar and had to give stroller away at the gate. I had about 4 hour layover in Doha and they have strollers at Doha airport, that's what I used during layover.