Controlling electricity flow via MIDI to power lamps
This is certainly feasible (which does not imply that it is feasible for YOU, that's your own judgment!).

There are two or three 'major' parts of your project:


driving the lamps. You did not state voltage or power, I assume you want mains powered lamps? If so and you don't have much electronics experience I'd advise to buy something ready made, with for instance 0-5V (isolated!!) or DMX input (is always isolated). 
programming the Arduino to interpret MIDI. This is not a big deal, but I am a programmer. If your programming is rusty this will be the interesting part.
programming the Arduino for driving the output to the lamps. This depends on the hardware interface to the lamps.


Blowing a fuse: switching a cold lamp on draws a short current peak. You might consider keeping the lamps always on at at least a minimum level. This will reduce peaks and extend your lamp's life. You did not specify the lamps you want to use, so I can't comment on possible fuse problems at the location.