I lived in Brisbane for 4 years while studying.
I am here in SG as a professional expat. Good place to get started or continue with career experience.
Before I address your questions, I think the first priority is to remind you that should talk to other expats in your field who are already working in SG. SG is a high pressure competitive workplace, not everyone can handle the 'Asian work ethic' particularly over the medium/longer term. Most expats burn out because of incompatible work culture.
I'll address your other questions: Cost of living in SG is not very high compared to other asian cosmopolitan cities. Housing/rent will be your biggest cost, but it can vary between $1000 for a single room with own bath, to $2500 for own small condo 2 bedroom unit. It depends on your needs and if you're here alone, don't mind sharing, etc. Finding housing is difficult unless you have a job here, and housing is mostly signed with 1 year rental terms minimum. As always, location and access have the biggest impact on rental costs.
Be prepared for 'selective' landlords, so you'll have a tough time finding housing unless you fit the demographic of young chinese single female professional who doesn't cook at home and is fine with sharing living space with the landlord uncle. Have a look at the local gumtree SG for rent listings to sample.
Connectivity: Gbit fibre internet is cheap, $60/month for 1000/500 unlimited. Mobile plans can be as cheap as you are willing to endure. I have a 4G plan with 3GB limit for $30/month.
Food: Food is cheap if you are willing and have time to cook at home (NB. many rental landlords have strange and strict limits on cooking at home). If you value your future health I would advise cooking at home at least some days of the week due to the health risks associated with eating out regularly.
Transport: Unless you have $$$, get used to taking public transit. For reference you can research the cost of owning and operating a car, it's insane (but warranted and rational; it keeps the poor off the road and traffic gridlock controlled). Public transit is excellent. My monthly usage is around $50 going from city to the suburbs for work.
Living: Singapore is a port/commercial shipping hub. You can buy almost anything here if you have the money. Prices will not be cheap but if you want something you'll likely find it here, or it's a cheap/quick flight to neighboring countries where you'll find what you need. I buy my computer parts from Amazon, cameras from Hong Kong, and Whiskey from Japan.
Outdoors: 3 words: smog, humidity, noise. The smog (seasonal peat bog fires from Indonesia) along with local vehicle exhaust is equivalent to smoking 3-5 cigs a day. When the smog hits, wear a N95 mask. I have an N95 air filtration fan at home.
humidity: some like it, some hate it. Some die from it while attempting the marathon. I would advise you visit SG before committing to long term stay here just to taste the weather.
noise: it's a small place, there is no escape from all types of noise pollution (vehicles, people, the background hum of all the aircon compressors). It is rare to find accessible tranquility in this town.
I used to bike centuries, but since moving here I've kept my cardio indoors due to the above reason, not to mention the roads are deadly for cyclists.
The good? Cheap flights to South East Asia and Asia Pacific. Hyperexposure to new peoples, cultures, foods, art and so forth that you won't easily find in a less global city.
Bottom line: Come visit, explore your career options here before committing.