A big +1 on what Graham Meek said above ^^^^^^
Only the bearing register surfaces and the R8 taper surface would usually be precision finished at the factory, probably ground. The area in the middle of the spindle could well have been left "rough machined" and not precision ground because is does not contact anything, It just sits there in space. So runout there is ok.
Graham's very handy suggestion of setting the spindle up in the four jaw and fixed steady so that both the bearing register surfaces are running dead dead true, then clock the taper is a much more stable setup than trying to rotate the spindle on V blocks that are not bolted down and clock the taper at the same time.
Then atfer this above test has shown the spindle is (hopefully) running true, when you get the new bearings and install them, I would set it up similarly with one bearing held in the chuck and the other in the fixed steady and then rotate the spindle and clock the R8 taper. It should run dead true if all is well.
Installing the new bearings is usually done using mild heat to expand them so they slip easily onto the spindle. Heating them up in a tin of oil over a flame is the traditional way. An electric heat gun can do the job too, with care. If you do have to gently tap the bearings onto the spindle, support spindle on block of wood and use a tube etc so you are tapping only on the inner beraing race and not transfering shock through the balls.
I would then do a final test with the spindle and bearings installed in the quill and lay the quill down or clamp it down to something or hold it in the four jaw and fixed steady again etc and clock the taper as the spindle turns in its bearings in situ. If runout mysteriously shows up at this stage, it must be due to bearing installation not being quite right somehow. Very unlikely though. But best to check anyway.
Definitely leave the press alone at this early stage!