Finding integer square root for large integers [find asymptotic time complexity]
Normally, you don't measure the complexity of a problem before designing an algorithm; instead, you measure the complexity of a particular algorithm.  So, before you get to measuring complexity, the first step is to come up with a specific algorithm.

When you have a specific algorithm in mind, one technique that's useful for analyzing the running time of divide-and-conquer algorithms is to write down a recurrence relation characterizing its running time, and then solve the recurrence.  You can often use the Master theorem to solve the recurrence.

To learn these techniques, see your favorite algorithms textbook, or read the following resources:


How to come up with the runtime of algorithms?
What is the difference between an algorithm, a language and a problem?
Solving or approximating recurrence relations for sequences of numbers
Is there a system behind the magic of algorithm analysis?