I use Notepad ++ and whenever I load it, it sometimes tells me a new update is available, giving me the opportunity to update it.
Is there a way to do this with the wallet as well?
I remember asking this years ago when I was a noob back in the Bitcoin-qt days, and I remember that the answer I got was that the software didn't auto-update in purpose, in order to make it optional for people to do so. In case there are big changes in the system, they don't want to push this automatically into the network, the update must be opt-in, and so this is why you don't see a prompt that tells you to update, you must manually check bitcoin.org for updates.
That is actually quite good reason. Bitcoin is decentralized, and bitcoin.org (or anything else) is not an authority that should have power to push their update automatically (or semi-automatically for that matter). All changes to the system should be originated from the system and not made upon merely consent, but upon user's own wish to upgrade.