Can a Nikon D3100 shoot fast enough to take a picture similar to this?

is there a certain setting it needs to be on?

Yes. Put the camera on shutter priority, put enough light on the subject, and try 1/1000th of a second, and let the aperture set accordingly.

just to put it in perspective, 1/4000th of a second is fast enough to stop helicopter blades without any blur at all, on a sunny day.

Yes, but you need a lot of light to get your shutter speeds up. That shot was most likely done in a studio enviroment with very powerful studio lights.

There isn't really any specific setting that your camera has to be in, except making sure that your shutter speeds are fast enough (at least 1/500th of a second). Something like this requires trial-and-error; I doubt that person got that shot on his or her first try.

I took the trouble of lookingg at that picture you posted and that was taken with a shutter speed of 2 seconds. The whole tirck is that you let your lens open and freeze the drops with your flash.

http://www.diyphotography.net/the-comprehensive-water-drop-photography-guide

I would consider its a good camera

Nikon D3100 14.2MP Digital SLR Camera
* 14.2-megapixel DX-format CMOS image sensor;
* 3-inch monitor with One-Touch Live View shooting and movie capture
* Full 1080p HD Cinematic Video with full-time autofocus and sound
* Easy-To-Use Nikon Guide Mode with intuitive controls and on-board assistance