Just to add my perspective...
I grew up in Sydney and moved to Melbourne for work at the beginning of 1993 - and I am still here. That being said, I am often in Sydney (family, work and socially). Going to Sydney in WHY is no trouble - it is only about a 1-1.5 hour flight.
I lived in North Carlton when I first moved to Melbourne. I don't really know what the apartment scene is like, but the suburb is very nice. Drummond and Canning Streets are great, as is Rathdowne Street (even though it is a bit of a main road, an apartment not on the street would be great).
I have also lived in Middle Park (one of the inner bayside suburbs). It is great too and conveniently located for St Kilda, Southgate and the CBD.
(Other Melbourne places I have lived - Coburg (too far out for you by the sounds of things (7km), but multicultural and gritty. I loved it at the particular time in my life), South Yarra (I am not a fan of the south side of the Yarra, but this is purely a preference thing - I much prefer the inner north which is closer to what I grew up with in Sydney in inner city Balmain) and Clifton Hill (a wonderful, eccentric little suburb in the inner north east. Probably a shade too far from the CBD for you, and you do need a car if you live there. It is now also shockingly expensive).
Since returning from an expat stint in London at the end of 2001, I have lived in the CBD. I love it the most of all of the places I have lived in Melbourne. The CBD has really developed as a place to live. There is a proper supermarket on Lonsdale Street in the New Queen Victoria Building development, the Queen Victoria Market (biggest and best in Melbourne - foodie paradise) is about 10 mins walk up Elizabeth Street, all of the big department stores are handy (and open 7 days/week) and there are lots of great bars (all within walking, or staggering, distance), restaurants (Little Bourke St for Chinese food, but you MUST get locals knowledge as many of them are for the punters - here are some starters - Shark Fin House for dim sum, and Crystal Jade (this is tucked away in one of the side alleys off Little Bourke St) for fresh seafood and non-gloopy proper Chinese food), coffee shops (again, locals knowledge required to avoid the tourist traps), cinemas, museums, the Arts Centre etc etc all within walking distance.
Advice on getting an inner city apartment: some buildings are better than others. It will really pay to do your research on this one. There is also an oversupply of CBD apartments at the moment and rents are falling. That being said, you have to pay for quality. When I first moved into the CBD I looked at very many apartments. I am a keen cook so needed a decent kitchen. This was a challenge, as many of the newer developments seem to be aimed at people who never go into a kitchen. What I settled on wasn't ideal, but it works for the time being.
Do you need a car? To play golf, yes. The golf courses are all some distance away from the CBD. Royal Melbourne is said to be THE golf course and club. If you are a member of an overseas recognised golf club, keep your membership identification card current and get a letter of introduction written from your home club to play there as a visitor. Becoming a member though takes 10-12 years and plenty of cash, although it might be easier for ladies.
The weather: this is a vexed subject. First, don't listen to anyone from Sydney who has never lived in Melbourne. Secondly, what they say about extreme variability is sometimes true. That being said, rainfall has been below average for the last 8 years or so and things have not been as bleak on the weather front as they were when I first moved here (whether the city runs out of water in the next couple of years is a serious issue at the moment though).
In summary, Melbourne is a lovely place to live and certainly better than Sydney in a number of ways (and I say that as a former Sydney resident and someone who goes there at least twice a month). If you are not disadvantaged from a business perspective by not being in Sydney (which is now the business capital of Australia), then you are sure to love living in Melbourne. If you would like more information, please feel free to PM me and I will send a number on which you can speak to me.