My wife purchased a home based printing business this past year, which included some old equipment and the current customer base. Can I deduct the initial capital investment to purchase the business as start-up cost?

These are your cost to purchase a business. These are not start-up cost. Also start-up costs must be amortized.

You can deduct some of your capital expenses under section 179 deduction.

Can I estimate the value of the equipment and software, even though it was not specified in the receipt of the total purchase price of the business. The items were listed, but there was just one total purchase price of all the included items.
Thanks for your help.

There is always a breakdown of the total payment. Check with the Escrow company or real estate agent or from the seller what they are filing on their tax return.

Can I estimate the value of the equipment and software, even though it was not specified in the receipt of the total purchase price of the business. The items were listed, but there was just one total purchase price of all the included items. ---- If the value of each item was not listed in the purchase documents, you need to estimate the value for each item. The estimates should be based on fair market value.