Arduino Programming Language Capabilities

Is Arduino capable of running sophisticated code like one which is able to tell the sunrise and sunset for everyday? I want to make a DC motor activate do to this code. http://www.esrl.noaa.gov/gmd/grad/solcalc/main.js

Yes

Once you've got a real time clock with month and day (and you'll probably want an RTC module) you can simply use a table of sunrise/sunset times for your location, and store that table in memory.

If you want to minimize memory usage, you can use weekly values (or even monthly) and then interpolate for the daily changes (if you need that accuracy). The day-to-day differences are greatest around the 1st day of spring and the 1st day of fall, so that's where any interpolation/approximation error will be greatest.