What is an average hourly rate for a business lawyer to conduct startup activities for a business client? -

In addition, would this type of lawyer also work on contracts for a business client and would the rates be the same?

In the US, business lawyers will change around $300 per billable hour. They will be proficient at most/all levels of business legal support and services. Choose one wisely however... Do your homework and seek references before you engage a lawyer.