Have a look at RAPI.EnumerateFiles(). It returns an array of FileInformation.
Therein you'll find the FileAttributes with 16 for example indicating a directory.
Assuming you have your RAPI object instanced as rapi and it is connected,
this will give you the names of all directories starting with 'backup_':
var backupDirs = from dir in rapi.EnumerateFiles("*")
where (FileAttributes)dir.FileAttributes == FileAttributes.Directory
Of course you don't need to use linq ;)
Hope it helps.