If you haven't gotten a letter yet, I wouldn't necessarily expect one at all. With the program downsizing, the amount of information most of us recieve about what's going on is negligent and the amount of schools/duties we need to cover can be sort of insane. Your predesscessor could be busy, in the dark, or just plain lazy.I disagree with this assessment as the program has been downsizing for years, ALTs have always been poorly updated, and most predecessors are anxious to contact their successors (if only to sell them the stuff they leave behind).
There's a good chance that you will be contacted as it is not at all uncommon for the schools to notify your predecessor of your contact info with less than a month until your scheduled arrival. At this point someone from your school or board of education should have contacted you, and they are someone you can ask about setup money. Most BOE's are familiar with having to help their new arrivals setup and may be willing to furnish you with a loan.
Typical first month costs include:
Key Money/Security Deposit/First and Last Months Rent: up to 300,000 yen
Household items: up to 50,000
Cell Phone: typical phones cost around 20,000 yen, some plans require payment up front for the phone
Parties (welcome and otherwise): figure around 5000 yen per
Of course you may get lucky and have everything provided for you. Check with your supervisor or predecessor.