What the Apollo flight computers did is essentially compute what is called a Kalman Filter. In control theory, the Kalman Filter is the simplest form of what is called state estimation. What the Kalman Filter does is it takes the current position and velocity and says "this is where the rocket is heading" and compares it to a pre-computed trajectory, or "where the rocket should be heading".

It then outputs what is called a Kalman gain: this is a vector where each component is a number that tell one of the controllers on the rocket (gyroscopes, fins, engine gymbals, etc) how to adjust to put the rocket on the pre-computed trajectory.

The actual mathematical operation going on is a least-squares estimation. This is basically about as complex as computing a bunch of standard deviations - it involves a few additions, mutiplications and square roots each cycle. You do not need many cycles per second to maintain accurate trajectory following.

Now back to the computer: this was not a general-purpose computer like your PC, or even a general-purpose calculator like one you can buy in the dollar store. It was literally build to do one thing and one thing only: compute Kalman gains based on a bunch of sensor inputs. Every input goes through a bunch of transistor groups like adders and bitshifters and whatnot in the exact order that you would add and bitshift and whatnot in order to compute a Kalman gain. Once you boil it down to this, it becomes clear that you only need to figure out how many transistors it takes per input to compute this efficiently. The answer is not many.

If you are a very masochistic person, you could go buy a few bags of transistors at RadioShack and make your own rudimentary Apollo-style computer using that era's technology.

Now, Im not trying to say this is easy to do. It is not. It takes some very smart people to do this well. But it is perfectly feasible given a blank cheque with rudimentary tech.

To put this point home, you could build the same computer with vacuum tubes. It would be huge and heavy and require probably a separate module on the rocket, and it would most definitely fail due to heat stress in space, but it would have the power to compute Kalman gains.

Tldr: flying rockets takes little computer power.