It's a tiny "crack" in the tire maybe a centimeter wide and less than an inch long. Can I fix it or do I need to buy a new tire and what size do I get?
Is that 1cm x 1in, or 1mmx1in?
A cm wide crack is pretty large for a front tire. A mm would be reasonable for moderate to severe weather checking.
If it is still holding air, it can probably be patched, assuming the problem is with a tube. However, if it has very bad weather checking, I'd probably go ahead and get new tires.
For vehicle tires, I've been watching Craigslist for narrow 16" tires for some time for a pickup project. They do show up occasionally, at least around here. But, not that frequently. Perhaps I'll see them once a year or so. I think I did eventually find a pair of 6.50x16, and four 6.00x16 tires, for about $25 each. Otherwise, new vehicle tires can be rather spendy, and not necessary.
There is also a temporary tire, T155/90D16, 60PSI. You might be able to score a couple very very cheap at your local junkyard. Perhaps $5 to $10. I certainly would consider them for occasional light field & yard use with a tractor, but not heavy brush use, or lots of tractor use. Perhaps as a temporary stop-gap measure.
Anyway, if you have patience, and want car tires, just keep your eyes open. If you need AG tires, then you may also find them on Craigslist, or at your closest Tractor Breaker (also probably has rims). But, you may be best off just going to a local tire store and buying new. You may have troubles finding good, super-narrow ag tires.
Ahhh... here are some numbers.
For "new" vehicle tires, 6.50-16, $124 to $143
Tornel Hwy 6 Ply - 650-16 - Military & Truck - Tire Applications
STA Super Transport - 650-16 - Truck - Popular Tire Applications
6.00-16 $181
Firestone Knobby Truck Tread Blackwall - 600-16 - Truck - Popular Tire Applications
New Tractor tires are much cheaper, and perhaps easier to find.
5.50-16, $60
6.00-16, $62
FarmTireStore.com
Anyway, talk to your local tire store before buying them online. They can probably source them for you at a similar cost, and they will much prefer mounting their own tires.
I got a 4.00-9 tire from my local store. It took them about 1 day to find it.