Update script fails due to Service capacity
Attempt to answer your questions ...


  Has anyone seen this one before? If so how do I solve it?


I've never seen this error before. But what I don't see anywhere in your question is that you FIRST disabled all modules, prior to upgrading Drupal core. And that is problably where your error originates from.

So if you have not done that, try doing so now as a first step. Note each of the modules you're disabling, so that afterwards you know which one to re-enable again. Note "all" is more then just the 18 modules you mentioned that needed updating (so also those modules which do not require any update).

After all these modules are disabled, try to upgrade Drupal Core. I'd expect that upgrading core then should work just fine.


  If not, can I run the other 18 pending updates somehow?


I would NOT try to do that. Instead first make sure core is upgraded, as explained in my prior item.

After core upgrade completed fine, re-enable all modules again that you disabled to complete the core upgrade. From then on the 18 pending updates should appear also again (of course). Then perform the update of each of those 18 modules, one by one.


  If not, how harmfull is it if I don't run update.php?


NOT running update.php is not something I would consider. In other words: if your Drupal installation shows messages that updates are pending, you better apply them ASAP.

Does this help somehow?

PS: worst case (if everything else goes wrong): restore from your last backup (you have that, right???). Then apply the steps I described above. I'd assume you would not run into similar errors anymore.