Thanks to Kasandra for this Frugal Baby Tip!
I can honestly say that flying solo with a toddler is one of my specialties. With the diapers, I would suggest sticking with cloth if it is a short flight AND you have the means to wash them while on your trip. However, if this is a very long flight, you may want to use Tushies Gel Free disposables (or if washing them while away is difficult).
Also, when I flew with my son at that age (he is 22 months now) I left the stroller (big pain as a carry on) and just carried him and his car seat (he had his own seat on the plane) and a backpack style diaper bag. I checked everything else. If you have a sling then I would HIGHLY recommend going that route since back carriers are usually more bulky.
Also, a backpack style diaper bag is also very useful on flying trips. Oh yes, and one more thing I found very effective in keeping Sean well behaved was this: I went to Toys R Us prior and bought him about $20-$30 worth of small toys (that travel well) and kept them stashed away until the flight (when he saw them he was thrilled and forgot that he was stuck in a car seat for three hours).
I also brought him some special snacks that he usually doesn't get at home for the flight. This is a crucial point because kids act up really bad when they are bored (mine does anyway). It works because I usually get comments from other passengers on how well behaved he is (and believe me when I say that he is a very spirited child... to put it in the kindest terms possible [I love him the way he is, of course]). Good luck!!!