Yes. You can upgrade directly from BE 2010 R2 to R3. Your existing licence keys will work. Make sure you make a copy of the Data and Catalog directories before you do the upgrade, in case you need to revert.
You can download BE 2010 R3 from either the fileconnect site or
http://trialware.norton.com/files/fc/Backup_Exec_2010_13.0.5204_MultiPlatforms_Multilingual_DVD.iso
After your upgrade, do not forget to run LiveUpdate a couple of time to update it to SP4 and the latest hotfixes.
After you have upgraded to R3 SP4, push out the remote agent again.
BE 2010 is rather long in tooth. You should be thinking of upgrading to BE 15 which is the latest version. You would need a new set of licence keys for BE 15. The new keys are free if you are on a current maintenance contract.
