I'm looking for the cheapest transport option to reach the Muiderslot castle, from Amsterdam Central. Muiden is a good city to stay/sleep? I don't know if it's a good option stay in Muiden (to be near Muiden Castle) and also visit Amsterdam Centre.
Just did this today, by bike, from Amsterdam City Center.
Go to the Flevopark on Amsterdam's impeccable bike lanes, then take the scenic bridge (Nesciobrug) and pedal for a few kms, mostly right along the river, in a park with bunnies (Diemerzeedijk / Waterkeringpad).
You'll cross farms, sheep, friendly people. Muiden itself has a few nice cafes and is sailing oriented. The castle itself is beautiful and smallish, and they close at 4:30pm, so try to get in at least an hour before that. Don't miss the garden outside.
Also right nearby is an old fort (Westbatterij) with cannons, behind which there's a very small and cozy beach, you can swim and enjoy.
This is the route: Centraal to Muiderslot by bike
It takes about 50 minutes to get there and you have to change from subway to bus and then walk for a quarter of an hour. This wil cost you EUR 4.81 one way.
Muiden is too small a place to spend more than a couple of hours, so you'll probably want to return to Amsterdam.
Hiring a bicycle will cost you about 8 EURO. See http://www.amsterdam.info/transport/bike-rentals/ The bike ride will take about as much time as public transport.