Yeah, that must have been discouraging to hear, especially if you have limited understanding of how CPU/memory resources are allocated on a shared server. It's one of the "gotcha's" that often pop up when doing "heavyish" processing on a machine whose primary function is to serve webpages.
If you know nothing about the use of the *nix shell, and are not able/willing to learn, that kind of advice is not likely to be of much help (even if it is accurate). And that does encourage the assumption you make next.
... or are willing to learn a few new skills in that area, OR are willing to be creative in finding other approaches to the problem in addition to the advice support has given you.
One such approach might be to back-up your entire installation, program files, data files, and database (which, BTW you should always do anyway whenever attempting any type of an upgrade - stuff does go wrong with "release candidate" software, and even with "stable" versions ),use that data to install a local copy of your phpBB forum on your own computer, and do your upgrade there (where you control all the memory and CPU resources without impacting other users on your web server).
If the upgrade goes well, and works properly, you can then move that successfully upgraded installation from your computer back into your webspace and "rock on!".
This shouldn't require any *nix knowledge at all, though you will need to understand how phpBB works and know your way around MySQL,etc.
[color=#CC0000]Edited to add additional information[/color]
[color=#00CC00]Being curious, I did a quick google search, and it seems you are not the only one with such an issue, and I'm not the only one with the "do it locally" workaround idea. Check out this thread from the phpBB forums on using your own computer to do the upgrade. It seems upgrading this way is even easier than I suspected! [/color]
I think learning enough about the shell, a few s*nix commands, and "nice" in order to accomplish the upgrade would be a reasonable investment of time (as it should require about an hour of research time) for the anticipated return (you complete the upgrade, and those skills are likely to be useful in many future situations) but I understand YMMV on that.
That said, there is usually more than one way to do most anything like this; sometimes you just have to be creative.