You are mixing up definitions. Pi is computable, meaning we can calculate one by one its digits, but is not rational. If you can compute a number in a finite time, you can put it om the number line, but the only numbers that can be finitely computed are the rationals which are countable
You can put pi on a number line just fine, though. You can pick any range of the number line to look at, and determine pi's place to arbitrary precision in finite time.
With "uniformly random real between 0 and 1" you can't do that.
More relevantly to infinities, the program that spits out the digits of pi is finite. We can easily count through all the computable numbers, including every number you can specifically name, albeit in a weird order and with duplicates.
It's not a bijection, but if you have a computable number you can derive a natural number that represents its program, and if you have a natural (or rational) number you can derive a program that computes it.
