Strollers are free or at least they were last year but they sell out quick and it is first come first serve. My sister could not get one at 7:20 or so last year, they were gone! We had our own but when we went in around 4ish, there were a lot there and many people from the regular days park admission were returning them still - so I am sure if you get there by 6-6:30pm, it won't be a problem.
Someone asked me on a private message whether you needed to use a regular pass admission or a hopper to get in as early as 4pm and the answer is no. All you need are your paid MNSSSHP tickets. The turn stile CM's let you right in and even at 4ish, there were a good amount of DISer's going in. That gives you 2hrs of the park to do rides before it oficially closes and then they even leave one land open from 6-7pm for the MNSSHP guests while they set up the park and get rid of the daily people without MNSSHP tickets.
Unless you have an expiring hopper pass with an extra day, I wouldn't waste a parks admission the day of MNSSHP. We had to start getting ready around 2:30pm to be ready and on the bus stop by 3:15 which then got us to MK turnstiles right at 4pm. I can't imagine going to another park for only a few hours in the morning and using up a whole day on your hopper. I would just sleep in, stay at the pool or maybe even go to DTD to shop if you are itching to go somewhere prior to. Believe me, dd slept in that morning until 9pm and then rested in the room from 1-2pm after the pool and I honestly believe that is the only way she made it to midnight. MNSSHP is non stop and tired kids will not be fun and we saw a lot of them. She then slept until 10am the next morning so I was glad we didn't have any early ADR's or plans. We ended up doing the Epcot countries that day since they don't open until 11am.
HTH - have fun!!!