I think the recommendation to start with a more recent one and work backwards is a good idea, as the older ones might seem slow and old-fashioned to a 007 newbie.
My votes would be
Casino Royale a good intro for someone from the Bourne/ 24 era
Goldeneye - although relatively recent, it's very much a classic Bond, putting all the ingredients in (then shaking, not stirring, hohoho) and provides an interesting contrast with CR in how to relaunch the series and introduce a new Bond. One going for a totally new approach, the other picking the best bits and pieces from a classic formula
The Living Daylights - I personally prefer Dalton's second and final Bond movie, Licence to Kill, but TLD is more of a Bond movie; LTK may seem like a generic 80s action movie. I think a Dalton film should follow the Craig and Brosnan ones - he's in some ways like a prototype cross between those two incarnations of 007.
Goldeneye - the definitive Bond movie. Nuff said.
Dr No. End with the beginning, as Quentin Tarantino might say. I know a lot of people might vote for On Her Majesty's Secret Service, but I think your b/f ought to see Connery's introduction, his first 'the names Bond ... James Bond' and Ursula Andress rising from the sea.
If you can fit in one more, then go for OHMSS, for the sake of completion, try and get in a Moore effort, either The Spy Who Loved Me (in some ways as important a template for later 007 as Goldfinger was) or For Your Eyes Only. Probably better to go for TSWLM - if he's already seen a Craig and a Dalton movie, may as well give him some escapism instead of grit.