Melbourne is a fantastic city. I have lived in a number of Australian and OS cities, but Melbourne has a great mix of all the most desired attributes.
The sand belt, as QF WP suggested, does have a number of fantastic courses. Most are tightly held private courses. Dependant on your membership affiliations overseas, you may be able to play on them, but access is limited. At present a number of these courses are fast tracking members who meet certain criteria. Public access courses in Melbourne are of a good standard, and very well priced. Further afield, there are a number of excellent championship courses, just over one hours drive from the CBD.
Whilst the inner eastern suburbs of Toorak and Camberwell/Hawthorn are regarded highly, the inner bayside would be my recommendation. The areas of Port Melbourne, Albert Park and Middle Park offer great access to the CBD (<5km), are well serviced by public transport, are close to sporting faciilties and in the midst of great restaurants and cafes. Given the proximity to these facilities, it is surprisingly quiet. It all depends on your interests and where you will be conducting business.
A 2 bedroom renovated porperty will set you back between $350 - $450 per week, with brand new apartments in high quality high risies costing similar money. Meals are excellent value, with high quality main courses between $25 and $35 dollars. Cafes do coffee for $3 and cooked breakfasts for $10-14. The ubiquitous Big Mac is $3.40. 5-10 km in a cab is around $15.
The MEL-SYD dash is more than survivable in WHY. With flying time of around 54 minutes it is enough to watch the inflight news and have a coffee. It is well serviced by QF with 20+ flights per day, with QP's at both ends.