Since the decimal expansion of Pi is infinite and unpredictable, the number just to its left and just to its right would have to conform to Pi’s ever changing value if we are to suppose there are no gaps between successive numbers.
If that is so, then the two numbers flanking Pi, must also be flanked by such numbers that do not allow gaps. Logically, this process must continue for all numbers to the left and to the right of Pi.
If this is the case, then no other kind of number could ever lie to the left or right of Pi without creating a gap. How does Pi 'fit' on the number line?
The problem is that you could apply this type of reasoning to any number. On the real number line, it is impossible to define exactly which number is after another, as you have shown. We could get closer and closer rational approximations of $\pi$, but never $\pi$ exactly. The answer to your question is that numbers don't really 'fit' on the real number line as it is hard to even say what numbers like $\sqrt 2, e,\pi$ etc. really are, much less define the numbers that are perfectly adjacent to them.
Sidenote: as a result, the set of all real numbers is "uncountable".