Check this from metalink:
How to Convert a 32-bit Database to 64-bit Database on Linux? (Doc ID 341880.1)
And this might be helpful too:
Basically you take backup (there is many ways to do this) from 32-bit and restore it in new 64-bit system.
And after that run couple of scripts which you find from MOS doc: 341880.1
You can get 11gR1 64-bit download this way:
Previous Database Release Software
Oracle Database 10.2 and 11.1 are no longer available for download. The software is available as a media or FTP request for those customers who own a valid Oracle Database product license for any edition.
To request access to these releases, follow the instructions in Oracle Support Document 1071023.1 (Requesting Physical Shipment or Download URL for Software Media) from My Oracle Support.
NOTE: for Oracle Database 10.2, you should request 10.2.0.1 even if you want to install a later patch set. Once you install 10.2.0.1 you can then apply any 10.2 patch set. Similarly, for 11.1 request 18.104.22.168 which must be applied before installing 22.214.171.124. Patch sets can be downloaded from the Patches and Updates tab on My Oracle Support.
And about direct upgrade I have done that successfully for minor versions for example 126.96.36.199 to 188.8.131.52 so that might be possible also for 11.1. to 11.2 .