I haven't gone into YSNP through Billings, but I have through Bozeman. If prices are similar, I believe it is a bit closer to YSNP. Several of the majors fly to BZN. Road is 2 lane but was in good condition last time I was on it. There were a lot of flat, open sections so to that passing slower traffic was not a significant problem.
From my experience, summer crowds will have started to build by the time your are planning. Look at the occupancy for the YSNP facilities to get a sense of how crowded the park will be.
There are several other threads in this forum with a lot of discussion and advice on visiting YSNP that you may want to look through if you haven't already.
The suggestion from swei009 about a cooler is a good one. If you do fly into Bozeman, there is a Wal-Mart across the highway from the airport (or at least there was the last time I was there). Stop by, pick up a cooler, ice, and some food (the wood sticks the sell as replacements for mop, etc. handles make good hiking sticks). When you are ready to leave, either check the cooler, ship it back home with some of the extra stuff you picked up along the way, or offer it to a local charity.