At just under a mile, those *MAY* work. While the product description says "50 Mile" that is a huge marketing scheme. In the most perfect conditions of line of sight, with nothing in the way, and using a repeater, those radios might just barely get 50 miles. In the real world, you're still using low power, on UHF, with the built in antennas that loose most of the transmit power as heat (signal loss.)
Again, for under a mile, it will possibly work, but there's no way to be certain without testing them. You also mentioned the one place is uphill from your location. You could end up in a situation where you can hear the person on the hilltop, but they can't hear you. There are so many variables involved it's hard to say unless you test it out first.
These radios you linked to in your last post are approved for both FRS and GMRS, so you'd be legal. The FRS channels use 0.5 Watts of power. The FRS channels don't need a license, but at 0.5 Watts of power, you may not make even that 1 mile coverage you want. The description doesn't say how much power they put out on the GMRS channels, but if I had to guess, it's probably 4 Watts(?). At 4 Watts you stand a decent chance of getting your 1 mile coverage, maybe.
At $50 for the radio, and then $90 per person that needs a GMRS license (unless the other person(s) are your immediate family members), you'd still be over your budget of $100.
I'd say, if you're willing to gamble the $50 investment (or save the receipt and box for a return) you could test the radios first on the FRS channels to see if you could get the range you need. If you do, you won't need the license. If the FRS channels don't get the range you need, then it's up to you if you'd like to invest in the GMRS license and test further.
I just checked Verizon and T-Mobile's websites, Athens Ohio has 4G cellular coverage on both networks. While I applaud your effort to want to learn about radios, it seems like an awful lot of trouble, money, and testing to try and use two-way radios for something that a simple cell phone can accomplish. I mean, you're trying to reinvent the wheel here. You need a means of two way communication for under $100 that will cover a 1-3 mile range. That can be done with a prepaid cell phone and still leave enough money in your pocket to buy a cheeseburger and fries. Is there a specific reason that cellular phones (texting, calls, emails, etc.) wouldn't suffice for what you need?