That's not a common phrase in business English, since a Google search for the exact phrase finds no good hits. You need to (politely) ask your client to explain what he means.
Yes, this is sort-of an old-timey way of doing an invoice.
Though, it is slightly different. Essentially it's a pre-invoice quote / invoice. Like you noted, it's generally before work is done, whereas an invoice is usually after work is done.
Just write it out as a normal invoice or quote, but notate at the top that it's actually a Bill of Statement.
Have you asked your client what he means by bill of statement? That's the only way to guarantee that you'll give your client exactly what he needs. There's nothing wrong with that. Simply tell him that you are unfamiliar with the term and you want to make sure you provide him with all the information he needs.
I'm not familiar with the term bill of statement, but a Google search for bill of statement seems to indicate that it's the same thing as billing statement.
A bill for goods or services that collects several invoices from a given time, usually a month, into one document. The statement bill is presented to the customer, who is then responsible to pay for all goods or services listed on it. This is useful for repeat customers, who may purchase several goods and services from a business over a given period of time.Definition from financeglossary.net :
The monthly bill sent by a creditor. It gives a summary of activity on an account, including balance, purchases, payments, credits, and finance charges.Definition from creditorweb.com :
A billing statement is a summarized bill that is sent by a creditor to all its customers every month. The majority of billing statements sent out are sent by issuers of credit cards. However, most every type of lender will send, either electronically or via mail, a billing statement. A billing statement is a summary of all activity related to a specific account that took place during a previous 30-day cycle. The billing statement will show all new purchases and all payments received, as well as the current balance of the account, the finance charge that has been calculated, other fees, the available credit and the credit limit.Also see What's the difference between a bill (or invoice) and a statement?