Exchange redirected restore to different client
For the browse to work, you need to set the destination client to a Windows client that exists. It doesn't even have to have Exchange on it. NetBackup captures the Microsoft eseutil DLL in the backup. (That's legal because it's a redistributable binary.) NetBackup uses eseutil from the backup to access the backed up image. That way, the Exchange version exactly matches.
The actual restore may work. Try it. NetBackup harvests messages with their attributes and attachments from the backup image. It constructs new messages and attachments, and tells the Microsoft EWS service to save them in the target database. This will work so long as the message from your 2010 database doesn't have an attribute that is not supported (or has changed type) in your 2016 database.
If the restore does not work because of a database incompatibility, then you will need to stand up an Exchange 2010 server as the target of your restore. Then use Exchange tools to move the messages to your 2016 server. If it comes to that and server A is still available to power up, you could use Exchange tools to move the messages directly from A to B without the NetBackup restore step.