Note to OP - others had already commented on airport options but you might want to get help in getting an EZ-Pass to zip thru the toll plaza at the Queens-Bronx bridge crossings (RFK Bridge, Whitestone Bridge or T-Neck Bridge) as without it, the cash lane could add 10 to 20 minutes for your morning "commute" assuming that weather isn't playing tricks on you.
From major JFK car rental lots to the Van Wyck Expwy (Rt. 678 North) toward the Whitestone Bridge to the Bronx is typically about 15 to 20 minutes before 7:15 am as the morning rush only get worsen, if you are behind schedule with this, expect 30+ minutes to get up to the Whitestone crossing - going to the RFK or T-Neck Bridge (try to avoid it) will easily add extra 15 to 30 minutes as it meant a bad day to commute (wet roads or wind, traffic, accidents, construction spillovers, etc.)
Hutchinson Parkway, north of Co-Op City/City Island (just before the Bronx & Westchester County line) usually started slowing down with backup traffic as it switch from 3 lanes to 2 lanes, and typically moving along in bumper to bumper, stop-n-go fashion to the Cross County exits (going toward the Saw Mills/Taconic Parkways) as I occassionally do this route in the morning from Queens to eastern CT. Delays for this stretch can vary from 10 to 25 minutes due to traffic, more time if there's an accident and/or mixed with poor weather/wet roads, etc. Using Rt. 684 is definitely the best option these days vs. the Saw Mills (consider Rt. 95 North to Rt. 287 West as alternate plan B instead of the Hutchson, EZ-Pass Toll will speed you thru northbound as it's collected one-way only.)
If your fly into LGA instead, you can opt for either RFK Bridge to the I-87 North (Major Deegan Expwy) or the Whitestone Bridge, depending on local road conditions that morning and keep going to get out of commuter traffic coming toward NYC.
To make this work for JFK or LGA, your red-eye need to land no later than 6 am so that you can be at your car rental in 30 minutes or less, and hitting the highway by 6:45 am so that you can be out of the Bronx by 7:30 am - as very often, a few extra minutes will result in a substantial delay.
I would allocate about 2 hours and 30 minutes for JFK to Fishkill, NY (subtract 15 minutes from LGA) to allow for the random nature of metro NY traffic bottlenecks. It will most likely be closer to 2 hours (90 min. is very optimistic & unlikely) and once you are north of Rt. 287 in Westchester, your commute should mostly be at/above posted speed limits and moving with the pace of traffic (just be careful with NYSP and their laser traffic enforcement.)