I recently reinstalled my OS, and I had an old backup of my wallet.
I am wondering if my newer addresses and transactions will appear when my client has finished downloading the blockchain?
The most important thing is the private keys, the wallet file has a buffer of 100 keys so unless you created more than 100 addresses (including change addresses) since the backup you should be good. If you did you could be in a serious problem; wallet backups should be done periodically.
Transaction information can be obtained from the blockchain. It may help to use the -rescan parameter when starting Bitcoin the first time.
Any sending addresses in the address book, and labels you gave to your receiving addresses, which were done after the backup, will be lost.