I have a 15 amp breaker with 2 outlets & a fluorescent ceiling fixture on it. Can I run a toaster oven rated 1800 watts if nothing else is running on the other outlet?

As a practical matter, no. 1800 watts at 120 volts is 15 amps. However voltage varies. At 118 volts 1800 watts is 15.25 amps, at 124 volts it is 14.5 amps.
If the toaster doesn't trip the breaker today it probably will tomorrow.

I had an ancient condo once that had just one 20 amp fuse for the entire place. I managed to never blow the fuse. Fridge (900w), air conditioner (6.5 amps), microwave (1300w), computer, TV, toaster (900w), lights. Didn't use the AC, microwave, and toaster at the same time as each other.

Doesn't really apply to this post, sorry, just wanted to sort of brag about it.

Yes, you can. The breaker will not trip at 15.5 amps, probably ever. It will not likely trip at 17 or 18A in the few minutes a toaster oven will be running.
Of course it's not ideal, but it will work fine.

just for fun, the breaker trip curve on a Square D "QO" 20 ampere breaker crosses the 20 ampere trip line in 8.3 minutes...(500 seconds)

That should be plenty of time to ride through a toaster oven.