We cannot write the code for you, but we can give you some hints as to how to accomplish the result.

I have the following idea:

To make the number a palindrome, you need to make the rightmost bit match the leftmost bit, the second bit on the right match the second on the left... repeat for the third, fourth, etc., until you reach the center of the binary number. You wish to change
the bits on the right and not those on the left, since the former will require a smaller number of increments or decrements. The number of increments or decrements is 1 for bit 0, 3 for bit 1, 7 for bit 2, etc., i.e., 2^N-1. This gives you a quick way to calculate
the number of increments or decrements needed to make the number a palindrome.

However, I think that coding this in 20 minutes (as you are required) is a bit optimistic. Unless you are extremely proficient at writing code and manipulating binary numbers, coding this correctly and accounting for all possibilities, and testing it
properly, is likely to require significantly more time.