That would be fine for gaming at such a relatively low resolution. I strongly recommend making sure the card you get has two GigaBytes of graphics memory. Newer games can easily run out even at lower resolutions, and maybe you'll get a new monitor as well soon enough.
That one is on sale for $145 and will max any game at your resolution and many at 1080p should you upgrade your monitor in the future which I'd heavily recommend as 1360X768 is a terrible resolution. Keep your eyes peeled as these things have really been all over the place. They were on sale for $120 a few days ago.
If not than the R9 265 is the way to go.
There is a point where if you just spend a little more you'll get much more performance The R7 260 and lower don't offer very good price to performance.
With the fall in crypto mining of late you could pick up a 7950/70 for that kinda money. Just grab one that has warranty remaining. 2nd hand 7950's here in oz are <$120 at the moment.
I agree I would never buy a non-R9 card. Definitely upgrade that monitor as well ASAP. I went from that same res to 1080P and the difference is amazing.
A small roundup of 270x's http://us.hardware.info/reviews/5003/five-amd-radeon-r9-270x-tested-asus-msi-sapphire-and-xfx
Didnt look all that long but there is probably is a 270 roundup somewhere. At the end of the day the 'X' models are just higher clocked 270's (same core etc.)
I am a fan (excuse the pun) of the latest Gigabyte Windforce cooler models at a reasonable price then grab that (3 year warranty as well).
Overclocking wise its always down to the silicon lottery.
TLDR: Grab the cheapest card with the best warranty. Just read reviews of the chosen card first for noise levels etc.