If you can get the restore, the statistics and fragmentation will be the same after the restore (assuming you restore it to the same version of SQL Server). But you won't get any execution plans. Those are stored in memory on the server. But, assuming your server has the same amount of memory, the same number CPUs and the same system settings, you should see mostly identical plans get generated.

What you would really like to get to make your performance tuning better is a collection of the calls to the queries on that other server. Since you're on 2008, you can use trace events to set that up. You should be able to send them a script that will capture the information out to files. You can then consume it back in your own office to know which queries are the longest running, use the most resources, are called most frequently, etc.

You want the full cycle of performance tuning? For that one, go to my book.