You forget rockets were being fired at London 25 years before the moon landings, and von Braun didn't have any computers figure it all out. When you think about it, so much was achieved with nothing more than a pencil and a slide rule, and in some ways computers have made us lazy I think.
Yes, and with the top mathematicians and scientists of the age!

And the maths isn't *that* difficult.
Convoluted and time-consuming, yes, but nothing that couldn't be worked out in advance over the course of months.
The Earth and moon aren't suddenly going to change direction or distance - they keep plodding on at the same velocity - the only last minute calculations would be due to, eg, wind velocity as the apollo left the atmosphere.
Perhaps a couple of hundred miles of drift on a journey of 250,000 (each way).
A ZX Spectrum was MORE than powerful enough to calculate the entire journey - in real time - cos I wrote a programme to do so for my A level computing project.