To upgrade the Museum and get the art gallery, you will need to find Redd when he visits your Island. Make sure that you do not buy a fake piece of artwork to donate to Blathers. If you try to donate a fake, Blathers will not accept it and you will need to wait until Redd shows up again before you get another chance to buy a different piece of artwork.
Once you've bought a real piece of art and it has been inspected by Blathers, you can donate it to him. After you have donated one piece of real artwork, the art gallery will open in the Museum the next day.
To get Brewster you first need to meet the following conditions:
Once conditions are met you can talk to Blathers about the Cafe. You will see him thinking. He will then ask you to help find Brewster and give you a copy of Brewster’s Photo to keep.
To find Brewster you will have to take a Kapp’n Boat Tour. To go on one of the Kapp’n Boat Tours you need to talk to Kapp’n on your island’s Pier. You can only do one a day. So make sure to talk to Blathers first. It will also cost 1,000 Nook Miles.
You will find him hunting down some Gyroids — something he loves very dearly.
When you head back to your island go to the Museum and speak to Blathers to report your new findings! After a short conversation, Blathers will announce his intentions to build The Roost café.
Using the Amiibo Phone found in the bottom left of The Roost, you can also invite villagers and special characters of your choice into the café! Just interact with the phone and tap a matching Animal Crossing Amiibo Card.
Blathers now offers special posters to celebrate the completion of an exhibit!