Given the fact that he can shoot a ridiculous amount of webbing from each cartridge, they have to be quite dense and must have a substantial amount of weight. Given Spider-Man's strength, we can assume that he can carry a few cartridges without much difficulty and also fight with a cartridge on each of his wrists without much setback. Has it ever been mentioned in any comics or anywhere else how much each of his cartridges weigh?
Based on this article on the Marvel.com website, we can begin to rough out some numbers.
Dimensions: Based on a rough eyeball assessment, I'm assuming the cartridges are somewhere around 60mm x 60mm x 15mm in height, length and depth in order to fit into Spider Man's web-shooter without sticking out.
Materials: We know that the body of the cartridge is...
mainly [made from] nickel-plated annealed brass ... and sealed with a bronze cap which is silver soldered closed.
For ease of calculation, we'll assume that the whole cartridge is made of brass.
Weight: Given these constraints, the upper weight of the cartridge cannot be higher than 1 pound (0.45Kg). The cartridge also has a hollow chamber (containing the spider-string) which means that the overall weight must be slightly lower, possibly by as much as 75%.
Spiderman has a belt which contains approximately 20 of these cartridges (along with a hidden camera) which means that he's carrying approximately 20 pounds of weight (9kg) around with him. Although this may seem a non-trivial amount for you or I, since Spider Man possesses superhuman strength this weight may barely register with him.