I'll just add that if you are interested in the Van Gogh Museum you will need to book ahead on that. Go to their actual website and book thru there. They start booking about 3 months ahead and you can keep track on the ticketing site as to how full they are getting. They do sell out especially during high visitor times. They do not sell tickets on site, everything has to be booked online. They do not have an app, just book thru the website.
https://tickets.vangoghmuseum.com/en/tickets
If you also want to see the Rijksmuseum they are not as likely to book out since there is not a huge special exhibition coming up. You can also download this museum's app and book your tickets thru the app as well as use it as a guide within the museum. It is an excellent app for an excellent museum.
Anne Frank House is also a timed entry situation and you'd need to book that as soon as your date comes up for sale.
The last place I stayed in Amsterdam was De Ware Jacob Boutique Hotel which is near the Museumplein. However it's Sept prices look way over even your revised budget.