I am currently running Atahualpa 3.4.6 on WP 2.9.2.

I am going to update to Atahualpa 3.7.10 and WP 3.5.1
(I have your recommendations for updating from the Forum)
I did the update recently to several other site, but those were Atahualpa versions that allowed export of settings, so it went fine.

Since this version of Atahualpa does not allow export of settings,
is there some way of getting the settings from the older version to the new version
without doing a screen snap of every single screen, and then manually entering everything over from scratch? That will take forever?

Often you can often do a manual upgrade as outlined in the FAQ which allows you to switch back to your old version if things aren't quite right. The settings for Atahualpa are all in the database and usually the new install will just see them without having to import anything. Of course you should do a backup before doing anything "just in case". With a manual update you will need to copy images you have uploaded to the theme folder and changes a few path statements.

and if you upgrade to 3.4.9, you will be ale to export your settings...however (don't you hate howevers) due to WordPress theme requirements, the export/inport method used in 3.4.9 change at some other point.

But, as Larry said, you should be able have the two releases insealled and then you can export once you are at 3.7.10

and one other thing, the options are stored in the database so if you have a backup of the database (you did do a backup before updating) you have a copy of the settings.