I am a browser actually, stayed in both UK and SG (SG by birth). You seem like a cheerful chappie who is probably very excited by this opportunity; I couldn't help but register and reply to your question. OK, just my 2p worth and some tough love:
1. The package is not too great, tbh. You are looking at less than SGD3500, bearing in mind you would have to shoulder any exchange rate/comm to access your salary paid into your UK account. It is do-able but man, it would be tight and it might curtail your holiday ventures a bit. Observer made some very good points. Get thee hither to your nearest Citibank (there is a branch at Hanover Square) and register for an account. You could transfer money between banks easily.
2. Ensure that you have all the right information regarding the relocating costs. ie. flight, two weeks hotel, any shipping costs for even a few tea-cartons, any subsistence allowance or meal and transport reimbursement for the first two weeks etc. Is there any leeway for some sort of lump sum payment for relocation (even if it just a couple of hundreds of £). Moving countries is expensive business. At the mo, I think they are using your goodwill (and enthusiasm for this move) not to hand over the squids.
3. Singapore is actually more expensive to live in than London. Basic costs are higher. You can't get 50p loaf of bread, 99p bag of fresh salad or a £1.99 Veg subway nor could you get Tesco Value toiletries or generic medicines. If you are rather tall or a funny size, clothes can be a nightmare to find.
OK, now Singapore -
Have they figure what sort of work permit or pass you would be on and will they be applying for this for you?
On your budget, I think we need to discount any apartment or condos
which you would be looking at min. $2000 a month (without utilities). The option for you is a room in a shared house.
If you strike it lucky, you may get a room with ensuite, sharing kitchen and living room with a number of other expats in a condominium
with facilities. That is the best scenario for your situation, I think. Not so great, is a room in a family house in the older estates such as Ghi Moh or Dover. These can be "council estate" type like although rest assured, it would not be that bad. No druggies on the stairwells or grafitti. However, steel yourself for the worst. Ensuite or shared bathrooms are wet shower rooms instead of a bath.
The best move is for you to look at as many as you could in the short space of time. Check whether you can do light cooking, check what sort of contribution you need to make for utilities.
Most importantly, check what you need to pay upfront in terms of deposit and agent fees. You know that in London there are usually 4-6 weeks deposit + agent fees. Quite similar to Singapore although agent fees could be a month's rent upfront. Make sure you have the cash for this.
Like the earlier posters, get an EZlink card when you arrive. It is similar to an Oyster Card although really the discount is not that great, compared to Oyster. It just saves you the hassle. It would be great if you could walk to work but seriously, it would probably be rather hot and humid, unless you walk really early in the morning or late at night. I could leg it easily from Notting Hill to Soho in London weather but would be gasping if I do the same distance in Singapore's humidity.
Vegetarian Asian food would be easy to get in Singapore although it would be an adjustment to your taste buds. I think you would do fine there. Most vegetarian food would be of rice/noodles and stirfry/curried/stewed veg. If you are looking for a really fresh salad, you may find it a little hard or expensive.
Get health insurance sorted soon. When you arrive here, check what is the medical benefits like locally. If the local office have some sort of medical policy in place, ask whether you can join it. Usually, there is some sort of qualifying period (3 mths or so) but within that period, you can visit a doctor and get the costs reimbursed. Check this with the local company in SG, not through your UK branch.
In SG, doctor fees can be high and get this, they sell/dispense medicines at their clinics. There is a tendency to oversubscribe medicine (antibiotics seems to be a favourite). Coming for UK, you could do the self-medication route especially for things like colds and fever and just get the medicine from pharmacies.
Good luck for your move. Seriously, it will be tight but you are single and probably open for the experience.