The amount you will have to pay to outsource your bookkeeping needs to a professional will depend on numerous factors. On average, a person can expect to pay $500-$2,500 per month for an hourly bookkeeper. You also can look for a bookkeeper or bookkeeping firm that offers the option of a flat monthly rate. Some firms offer services as low as $95/month, like Remote Books Online.
If your business is on the smaller side with only one or two accounts, you will pay a much lower amount. If you have multiple payment providers you go through (like PayPal and Stripe) and multiple bank accounts and credit cards, you can expect to pay more. The number of accounts you have and the number of transactions those accounts see every month play a big factor in how much you will be paying for bookkeeping services.
Another thing to keep in mind is whether or not your business needs extra services. If you know you will need services such as sending and tracking invoices or catch-up work on books that have fallen behind, you will have to inquire what those services will cost. The more work there is to do, the more it will cost to have it all done.