Depends on the game, but older titles may not support the resolution. R7-240 does support the resolution so it will technically work. 4GB is about the minimum, however I don't think that GPU is really powerful enough to utilize it. Running a 4K display in 1920x1080 works out quite well so you could do that for a while as well. Though that GPU still isn't that good for that resolution.
When shopping for GPUs you generally want to stay away for DDR3 based cards and look for GDDR5 as well.