There is a shelter for thru-hikers right behind the ranger station at the state park (at the beginning of the Approach Trail), or if you get a shuttle to the top of the mountain, there are excellent camp sites right at the start on the top of Springer. So if you get an early flight, and have a shuttle lined up, it's possible to be camped on the trail that same day if nothing goes wrong
I'd want to be in Atlanta well before noon if I planned to be hiking that day. Yeah it's only a two hour drive to Springer but traffic can be a bear. Also consider that you'll need to get fuel after you land in Atlanta. Some of the shuttle drivers will stop at REI. (You can't carry or check any sort of fuel on a plane.) This can take some time. MARTA also takes a while if your shuttle is picking you up at the northernmost station.
Other options: there is the lodge at the state park, and also a campground. The lodge is not inexpensive, but it's nice and has a good view. Both are right on the approach trail.
In years past I'd enthusiastically recommend the Hiker Hostel to spend the night before beginning a hike -- it was a great deal for the shuttle, a place to stay, a great breakfast, and a ride to the trail, and it would give hikers some breathing room. I thoroughly enjoyed my stays there. But with the new ownership they've changed the rates and I'd want to see some reviews from this year's hikers.
Do think about avoiding April 1st as a start date - it's *very* popular and the trail will be *very* crowded, both that day, and basically every night for the first couple of weeks since the majority of hikers will hike the same distance each day (to the next shelter.) Starting in the middle of the week on a date that doesn't mean anything (e.g. "April Fools Day" "March Forth" or the first day of spring) would make it easier, even though there will still be plenty of people on the trail.
Good luck and happy trails