Arizona Stagecoach runs daily between Phoenix and Tucson (public shuttle van, not hiker-specific). Or you might consider flying into Tucson rather than Phoenix.
Greyhound bus runs between Tucson and points east along Interstate 10 (and of course between Phoenix and Tucson along the same highway). At one time in the past they made a regular stop in Benson, although you may be limited to Wilcox now, which wouldn’t be convenient at all.
Enterprise rental car has the “We’ll pick you up” motto. Many medium-sized towns have an Enterprise office. Check Sierra Vista, which is the nearest town to the southern terminus. Then see whether the Tucson airport location will allow for a one-way rental to SV. If that checks out, then call SV to see about drop-off options. The best case scenario would probably be to drop off the vehicle at their SV facility, then request that they give you a ride “home” to the border. They may not be willing to go as far as Montezuma Pass, and frankly I wouldn’t even suggest it given it’s a switchbacking dirt road. Instead, just shoot for the Coronado National Monument headquarters / visitors center, and then take Joe’s Canyon Trail from the parking lot there a couple of miles up to the Arizona Trail along the ridge.
Rental car companies will occasionally let you “abandon” a vehicle in the middle of nowhere, but in no case would that be included for free under the terms of your contract. For instance, it cost me $100 extra to ditch a car in Death Valley a few years ago and have the rental car company drive the 2-3 hours round-trip to retrieve it.
FWIW, I’ve heard Mt Lemmon has almost 4 feet of snow right now. Can anyone confirm? Unless the weather pattern shifts directions 180 degrees from its recent run of wet and chilly, this might be the year to begin later rather than earlier.
blisterfree