I keep a pail of  mild bleach water (about 1 tablespoon of *plain* bleach to
a gallon of water) and fill the bulb syringes and the 1-cup containers (that
I mix saline solution in) with the solution and keep them in the pail until
I them again.   Be sure to thoroughly rinse with fresh water before use.

Once a week, I fill the pail with white vinegar and soak the bulb syringes
and 1-cup containers for a few hours to remove mineral buildup.   White
vinegar (when not diluted with water) is also a great disinfectant.