OP, there are a couple of ways to measure if the mandrel and die bore are true to the axis of the machine.
This is what I do.
I have a 17 cm long piece of drill rod that fits snugly into the shell head holder slot on the mandrel. The rod was machined so that it would center in the mandrel when pushed all the way back.
With a magnetic based dial indicator button seated against the side of the press, with the dial set on "0", run the mandrel all the way up. This will give you a measurement of how far off true the set up is to the axis of the die hole bore.
The only other way is to measure the run out on seated bullets in the cases. Of course, this also needs a dial indicator and a special base set up to measure.
For most hunting purposes, the average press is accurate enough. It's only when you get anal about accuracy that such intricacies come into play.
If I were going to use a press to assemble match quality ammo, i would use the Forster press. The thing with the Forster set up is they require proprietary dies and shell head holders.
I'm just anal enough about such things, that I have a few match grade die sets made up for rifles I've built, using the same reamer that was used for cutting the chamber of the rifle. Even the shell head holders were custom made to be a very snug fit into the press mandrel. I will admit, this is over the top for most shooters and I only use it for a couple of precision long range rifles I shoot.
I got out of competitive shooting, other than the odd fun shoot, simply because I don't have the ability to know when to stop looking for the next best shooter. It consumed most of my time and became an obsession. I still shoot a lot and my obsession with accuracy still nudges me more than it should, but I'm actually having fun with the builds, rather than pulling my hair out while trying to figure out why a $700 barrel doesn't shoot that much better than a take off factory barrel.
If you aren't planning on being an accuracy freak, that new press you've ordered will very likely assemble cartridges with .003 in or less of bullet run out, which will be very suitable for most of the bullets that are presently available, which are better than ever since manufacturers started taking more care to keep jacket walls concentric and of consistent thickness all the way around.