Is it possible to find the position of a prime number online?

You can use the function prime_pi in Sage (http://sagemath.org), which is also available for free online at https://sagecell.sagemath.org. For example,

   sage: prime_pi(2011)
   305

Like Mathematica, Sage's prime_pi function is too slow to solve your problem above. It's also somewhat slower than Mathematica's still.

If you have access to Mathematica, PrimePi[x] will give you the number of primes less than x. Combined with PrimeQ, which verifies that x is indeed prime, will give you which prime number x is.

EDIT: I have no idea how long Mathematica would take (or if it could in fact compute it) for a number that high.

What you are looking for is the "Prime Counting Function". The closest thing to it you will find online is Wolfram Alpha:

http://www.wolframalpha.com/examples/PrimeNumbers.html