I am not a thru hiker but I know the areas he's hiking. The distance ~10 mi/day over the entire Tonto forest northbound to the base of Coconino Forest at the Mogollon Rim would take about two weeks give or take a couple days. You have to factor in that most thru hikers will take a zero day here and there to rest and the mileage can vary depending on the terrain. There is no way to know for sure but by May 11th your bro should be in the Vicinity of Strawberry, AZ. By mid June he'll easily be in Flagstaff or past it; either town would be a good place to meet up.
You definitely need to be in contact with him in advance; there is no way you will walk out on the trail and meet him by chance. Ask him to take a zero day or two up in Strawberry. That will be the best place for you two to meet up in the region of North Tonto / lower Coconino Forest along the AZT and is probably where he will be.
Here is a map of the vicinity:
https://caltopo.com/map.html#ll=34.4323 ... =mbt&a=mba
Notice the AZT heads east-northeast from Strawberry, sharing its designation with the Highline Trail. This is about a 13mi section of trail before the AZT turns back north. There are several trailheads along the Highline where you might be able to arrange a shuttle back to your car. however, getting to the the Washington Park TH where the AZT heads north might give a regular car trouble. It might be more worth your time to do half of that section with your brother and then turn around back to the car. Hope it all works out!