About the value: Check at www.antiquorum.com. They have a data base with all transactions of the last 1o years or so. There are several Ref. 1413 sold in the last years. Just type "Patek 1413" in the respective box of the welcome page and all auctioned pieces will show. You will be surprised about the value ... if it's in good condition.
You should send the watch to Patek Philippe Geneva by DHL and ask them for an offer for the overhaul. The result will please you, that's for sure. The watch will look like new. Ask them for an extract of the archives, which will cost about 100 USD, but it's well worth it! If ever you would have to sell the piece, it will be recognised as genuine and therefore more valuable.
An alligator strap is about USD 200. Make sure the AD also changes the buckle, which is rose gold as well!
Please show us some photos after you have received it back from Geneva.
Best regards
Eric09