Yes, it will be too cold for the beach. Expect crisp, fall-like weather. That being said, it is New England so anything can happen (although beach weather in october is rare).
Although out of season, I would think that most of the restaurants/shops would be open.
Enjoy

Thanks graypay.

Would love to hear more from others familiar with Block Island.

Shops and resaurants usually stay open until just after Columbus Day weekend (depends on staffing, many are college kids and return on the weekends for work), so you shouldn't find the island too shut down. The water will definitely be too cold for a swim but the beaches are certainly worth a walk. I'd suggest Mansion Beach, Town Beach and the walk along the beach at Mohegan Bluffs. In fact, depending how physical you are, walking and biking all around the island is the best way to see the real beauty that BI has to offer. The same can be done with a mo-ped, if you'd prefer. Suggested walks besides the beaches might include: Rodman's Hollow, Mohegan Bluffs, the Southeast Lighthouse, the Maze, and the North Light.

There are some great shops in and around Old Harbor where the regular ferry ports. New Harbor is where the high-speed ferries port and is mostly a marina. Not much going on there but good kayaking rentals.

If the weather's a little off, a visit to the goofy but still interesting BI Museum is worth it. And for restaurants, here's what I'd recommend:

For local color, go to the Beach Head or Ernie's or Bethany's Airport Diner. All great for bfast and/or lunch. Bethany's is the best after a big bike ride (lots of long hills). For good dinners I would recommend Winfield's, Eli's or the Atlantic Inn. I would avoid the restaurants right along Old Harbor, they tend to be tired (after high summer season) and a little cranky. Food is often less than good. Manisses is a bit tired, as well.

If you're really into biking, get the scoop on Black Rock. It's a vigorous ride (mountain biking) but just unbelievable, spectacular views.

And if you do go for bike rentals, they're all really the same price but not the same quality. We've had the best luck with the rental right on Old Harbor, right near the ferry dock.

Did I cover everything? One more suggestion: when in doubt, go the the BI Chamber of Commerce. While the staff might be a bit weary after a long summer they do have good information (you might have to ask the right question to get the right info) and can be very helpful. They often will give the most "popular" spot but when pressed, they might divulge the "best", too.

Have a ball on BI. We've been going there for nearly 15 years-got married there-know it and love it well.