How can i predict the next number of a long sequence of seemingly random numbers? How can I accurately predict the next random number between 1 and 999, given past results. I have no knowledge of the PRNG or algorithm used. 

How can I predict the next number of a long sequence of seemingly random numbers? Can I do it by hand, or should I use some kind of software to help find patterns from long strings of data?
This sounds like a lot of work.

I would break this down into parts.  First, I would try to identify the random number generator being used.  From that, my goal would be to determine the starting seed, which would let me accurately predict future results based on past results.  

I'd start with the NIST Random Number Generation statistical tests    In them, they offer 15 statistical tests for a random number generator.  I'd run the sample data through this battery of tests to see if it's actually random or not.  Important: if the tests reveal it is random, that could be evidence that a cryptographically secure random number generator was used to produce them.  I would stop searching as I would consider the problem unsolvable at that point.

I'd then test each of the random number generators that the original source might have used, such as rand(), and see if they reveal a pattern similar to that output of the mystery algorithm.  

Once a potential random number generator is identified, I'd turn to published papers to see what kinds of analysis and reversals are known for it.