The answer to the bottleneck question is yes and no.
First of all, let's consider what's important:
The graphics card processes pixels, and that's all it really knows about the monitor. Therefore, the pixel count is what matters. If you have a 1080p monitor (1920x1080), you will have a larger pixel count. If you have what companies advertise as a HD monitor (1366x768), you have a smaller pixel count.
The only real influence you'll see with larger pixel count monitors is that games run at slower frame rates. Seeing as you got a new graphics card, I'm going to assume it's not going to lag on the OS user interface.
Therefore, the only real bottleneck you'll notice (if at all) is that your games run at too high frame rates. This is where the yes and no come in. That can be a good thing, but also bad because you really don't notice the difference anymore past ~30 fps (some will argue up to 60 fps, but the idea here is that a really good graphics card with a low pixel count monitor will lead to results consistently above 100 fps).
This isn't really bottle-necking since nothing is plateauing to a maximum here, but it's buying something that you probably spent too much on. To check that, just run stuff like the heaven benchmark:
http://unigine.com/products/heaven/
If your frame rates are coming out high, you can just get a higher pixel-count monitor without sacrificing anything. Or keep the higher frame rate. As I said, it's technically not a bad thing.
Regarding the DVI/VGA cable, I'm not 100% sure, but really the only thing you need to make sure is that your GPU ports and monitor ports match up, or you'll need a converter. These are relatively cheap online. DVI to VGA or VGA to DVI.
And to answer your other question, yes you can just install the drivers and connect the monitor. These things should be plug and play.