My Bullit doesn't have a spare tire, your Bullit doesn't have a spare tire. What's the best way to deal with a flat, other than to have a spare, which I'm going to acquire shortly.
I've had quite a bit of experience with flat tires and the consequences, having worked for a Goodyear dealer in the past. Firstly, if at all possible, I would suggest you do not use the tire "slime" that comes with the car. It is very hard to clean out, can damage the rim, and under high heat can be, for lack of a more subtle word, "explosive." It's not flammable, per se, but I've seen tires blow off the rim when left in the sun after slime injection. Pretty much your tire will be ruined if you use it. If you do use it make sure you tell the tech whose fixing your tire that it was slimed. I had a tire blow off the rim in close quarters and a person can be injured. It makes a big bang!
Better alternative, buy yourself a plug kit, with the two tools and a strip of what looks like "turds." You pull the nail out, one tool opens up the hole then you push the "turd" into the hole with the second tool after coating it with rubber cement from the kit. Them pump up your tire. The plug repair is OK, and will get you home, but your tire's speed rating is reduced, because your are disturbing the steel cords and tire plies. Not the proper way for a permanent fix. More on that shortly.
Do not drive on a flat tire, you will ruin it in a short time. It may look fine, but when you take it off the rim, you will likely find a handful of rubber bits rolling around inside. I've had customers who got a flat, drove it off the highway, had a plug put in the tire and re-inflated, only for the tire to blow out shortly after. If you've driven on a flat tire for any distance it's probably NFG.
Proper flat fix: If you get a flat and there's a nail in the tire and you can re-inflate, do so and head for the tire pros.
A proper fix involves removing the tire, pulling the nail and, after prepping the tire around the hole, using a special patch with a spikey rubber thing in the middle.( see attached) Some come in two pieces. You pull the tube part through the hole from the inside and then the patch is adhered to the inside of the tire, and the tube thing is trimmed off. This is a proper fix. Put this tire on the right rear, in case you have a problem later, at least you won't be standing out in traffic.
If you are in the boonies and get a flat, you do what you have to do to get home, but quick fixes are not permanent fixes, and a significant cash outlay may be unavoidable. Get a spare. Road side "assistance" or AAA will just drop you off somewhere "safe" and take your car to the nearest tire dealer.