You can wash vintage cloth accessories, just be careful. I have washed (amongst other things) a dirty Yoda cloak and brought it back up looking like new. I literally used washing detergent (a tiny amount
) in cold water (no reason to risk shrinkage!) and then gently rubbed the detergent and soapy water onto the cloak and scrunched up the cloak a few times, removing the stains under the water. Then repeat a few times in clean water to rinse out the soap, lay the cloak out flat to avoid creases, and allow it to dry naturally. Voila! A nice clean vintage Yoda cloak.
The problem you may have with a Lando General cloak / cape is that they often won't come off over the figure's head, and although the head can (I believe) be pulled off, I personally wouldn't want to risk doing that. Instead, I would accept the fact that the figure will get submerged in the water and simply wash the cloak as described above. After rinsing off the figure and cloak, I recommend using the edge of a piece of kitchen roll (because it's tougher than toilet paper) and push it into and around the neck, shoulder and hip joints to remove water. Pad the figure dry with more kitchen roll, then stretch the cape out flat and let it dry naturally.
Good luck, and perhaps show us all a before and after shot of your handiwork!