First day in Amsterdam

Give yourself the best gift; book the b and b for the night prior; letting your host know that you will be arriving very early the next morning in order to get into the room upon arrival. Not sure where you are traveling from; but we made that mistake once arriving in Dublin at 6am after a flight from the west coast. And now when we arrive at our hotel, having booked the night before; we take our luggage to our room, freshen up; go out and explore until we can't take it anymore; and feel free to return to our accommodations at anytime we please. Believe me, waiting until 3pm or 4pm for your room is not fun on day of arrival when you arrive so early in the morning.

If its raining book The Rijksmuseum, if you like history stuff u have never seen, blow your mind stuff,, this is the place
Don't miss this if your in AMS
can only book tickets online

“ The weather will likely be cold/rainy, ”

How do you know that? There is no rain forecasted for the next couple of days and weather forecasts beyond that are notoriously unreliable.
There are lots of things you can do once you’ve stored your luggage. Take a canal cruise to get your bearings of the city, visit a museum, “fly” over the Netherlands in This is Holland etc etc

The RS guide book has detialed directions for a self-guided Amsterdam City Walk that starts at Centraal Station and ends in Leidesplein. It is listed as taking about 3 hours including going into a couple churches. There are places like Dam Square where you can stop and sit for a while and people watch, or cafes along the way if you get tired or cold. And you can always get on a tram if you are tired walking. This walking tour is what I did on my jet-lagged arrival day, but in an afternoon, after checking into my hotel. I find that being outdoors and keeping moving helps me stay awake the first day. If you have umbrellas or raincoats, light rain needn't deter you. If the weather is nice, a canal boat ride might be a good first day activity. I agree about not doing museums, at least not big ones, when jetlagged

For us, the key is to stay outside as much as possible and not sit down. Once sitting, you realize you are wiped out from the flight. What about a walking tour? Those take a couple of hours. A canal tour is great too. I would not do one of the usual big boat ones, but one that is small and you engage with the captain. We did both on our last trip and the smaller one was hands down much more enjoyable. Jack is the captain, and is in his early 30's and very engaging, so that will definitely keep you awake. He has private as well as shared cruises. We did the shared but were the only 2 people that night, so it ended up being private:)
https://amsterdamprivateboattour.com/tours/

Do the booking thing as noted above so you can check in upon arrival. Waiting until 3p is such a drag when exhausted.

Though mikeliz has a great suggestion for canal cruise, I didn't do that and promptly fell asleep on one in the mid-afternoon on arrival day. I'd opt instead for a walk around outside as much as possible. Save museums for a well rested day when you can appreciate them.

It sounds like your B&B is not staffed full time? The suggestion to book the night before is a wise one, but if there's not anyone there midday to let you and your luggage in that simply won't work. But check with the host to see if you could arrange something, you can't be their first guests to arrive on a red eye flight. Otherwise consider booking a cheap-ish hotel with late checkout near the B&B or train station (is that where you plan to store your luggage?).