Does anyone know anything about the Days Inn in Fort Lauderdale? Do they have shuttle service to the Port of Miami?

My guess would be that you mean the Days Inn Ft Lauderdale/Hollywood South. If so, it would appear to be that there are no hotel shuttles from there to the Port of Miami based on the information on the Days Inn website.

They do offer transfers to Port Everglades for a fee.

However I think you're going to be hard pressed to find a hotel in Fort Lauderdale that offers transfers to Miami as it's a bit of a hike. If you stay in a nicer hotel, the concierge can probably arrange for a SuperShuttle like van to get you down to Miami from Fort Lauderdale, but it will probably cost you.

Miami is a bit of a challenge because there aren't too many reasonable hotels near that port that offer shuttles to the pier. Heck there aren't too many reasonable hotels in Miami in general...