We often drive from CT to Ft Myers straight through. My suggestion is to stay awake. That way, you just go to bed at the regular time and wake up the next day ready to go - and on the same time schedule. If he sleeps during the day, it may be very difficult to get back to a normal sleep schedule. The last time we did the drive, we left at around 7 pm and got to Ft. Myers at about 6 pm the next day - showered, had dinner, hung out for a while and went to bed at 9 pm. Woke up the next day just like any other day.
There were 2 of us driving - we did try to cat nap in the van, but really got very little sleep on the road. Just make sure you have something planned to do when you get there to keep him awake!
On the other hand, you may be able to check in early - lots of times they do have rooms ready, but I wouldn't count on that.