In order to clean a very old stuffed animal first you have to spend some time preparing. Consider how old the toy is and what materials it has been made out of. There are old stuffed animals that are stuffed with straw for example and for them a washing machine treatment is a big no-no.
A stuffed animal can also be so old, that it has to be handled with care in order not to fall apart, tear or something else. This also means you should not use a washing machine.
If there is too much dust built up on the toy, use a vacuum cleaner. Put it on a low power setting and use a brush attachment. Take your time and with slow, gentle motions vacuum the toy.
If the stuffed animal is too dirty, then use the classic baking soda treatment. After you are done again vacuum the toy well, to remove all of the excess baking soda that will get into the fur.
If the toy in question is in good condition, you can do a surface wash. This means that you should get a bowl with warm soapy water and a sponge. With the moist sponge and slow gentle motions, clean the surface of the toy. Do small areas at a time and with not too much water. Then clean the sponge in clear water and use it to remove the soap from the toy.
Finally, the drying. It is best to leave the toy to air dry on room temperature. You can put the toy close to a heater but not too close. If the old stuffed animal is in good condition, you can also leave it in the sun for a couple of hours. Don’t do it if the stuffie is too old or brittle though.