How do I clean the shutter?

To clean and inspect the shutter mechanism on a Fuji E900 camera involves stripping the lens down almost completely to get at the shutter. It is not a task for anyone to "have a go" unless you have already upgraded to another model. You will need to have a few tools to hand - 12 watt soldering iron, tri-wing screwdriver, small phillips or cross head screw driver, needle point tweezers and another camera to photograph each stage for reassembly. I have photographs of a stripped E900 which maybe of help to you. My email should be listed somewhere in my profile or on my tips n tricks for digital cameras under "THE ONLY WAY I KNOW HOW.." and was posted under Fuji's A345 but does cover most Fuji compacts. There are some parts to the lens which are not accessable due to the way it was manufactured and will break if you try to access but the shutter is accessable for cleaning.

It's not really a practical DIY proposition on your camera as the shutter is buried deep inside the camera and was never designed to be touched after installation.

This camera is a four year old model, so professional repairs will not make economic sense as you can buy a perfectly good used replacement for less than the cost of repairs.

As the camera is effectively a write-off, you may decide to have a go anyway: if so you'll need a set of cross-head precision screwdrivers, an anti-static wrist-strap or workmat, a good light source and a pair of fine-tipped forceps. You'll also need a lot of ingenuity as it's not always obvious how to dismantle that which was never meant to be dismantled!

Before attempting a repair which is most likely to end up in failure, consider selling it for spares or repairs. You won't get much, but it's still better than having a pile of parts which eventually get binned.