How do I open a Progress .bup file? I have a .bup file which is supposed to be a Progress database file. 
My question is, how can I open/restore it and look at the data?
You need to have Progress installed. In it you will find a command: prorest run from command line that will help you restore a previously backed up database. At least if it has been backed up using the built in back up command (probkup).

To make things easier for yourself you could start the Progress command line environment (it should be in the bin directory in your installation path). Simply run proenv to start that and then follow the steps below. If on Windows you will most likely find proenv in your Start menu. 

Basic syntax:

prorest [new database] [archive]


so for example

prorest c:\temp\mynewdatabase file.bup


After that you can connect to the database using Progress built in commands. You might find more help at:
Progress KnowledgeBase
and
Progress communities.

A specific description of the prorest command can be found here: Progress KnowledgeBase Article 99033