I'm wondering how may versions "skips" were involved with your "upgrade" if your site has been dormant for a year? WIth each WordPress upgrade, if there are changes to the database, the upgrade process includes a program to update the database. I don't know if each subsequent upgrade includes all the previous or not.
In a perfect world, the "one-click" insatller/upgrader copies the new code to the appropriate directory and send you instructions via email on how to "complete" the upgrade. Often this entails running a database upgrade program.
The definitive guide to upgrading WordPress is available on the WordPress Codex and, even though DreamHost did the "hard part" for you "automagically, you should still review that so you know what is involved.
If all that is needed to upgrade the database from your old version to the current version can be done in a single step (depends to some degree on how many versions you upgraded "through" and the sophistication of the upgrade script from WordPress), all you have to do is run the single database upgrade routine by browsing to:
... (assuming your WordPress blog lives at yourdomain.tld/wordpress). If your blog lives at "http://yourdomain.tld/", then you would run the upgrade script by browsing to:
After doing that, the database should be fine (unless you have to go back and do incremental upgrades to get to the current version - and I do not believe that you will need to do this).
In the WordPress Codex article linked above are other good points about things you could encounter in the upgrade process, including handling your cache and possible permalink regeneration.
I think you can handle it yourself, or with the help of a tech savvy friend. You can always post back here if you have a problem, and we can try to help you further.