It depends on how your Access tables are set up. I had a database that had multiple tables with a single key variable (so you could follow a single case through across multiple tables). Matching those multiple Access tables up into a single SPSS file was a nightmare. I eventually abandonded the effort. It was dreadful.
In my experience, it would be best to create a new table in Access that combines the necessary tables/fields. Then import that new table into SPSS.
If you run into the error message about having too many fields, make sure you "Compact and Repair" the database. The 255 field limit includes any fields you create and delete or create and change (meaning it counts those fields twice). Compacting will reset the field counts so that each field is only counted once.
If you still have more than 255 fields, then you're going to have to get creative about importing cases. Hopefully, you won't have to deal with that issue.