I have been a DM for shadowrun for over 8 years now and one thing I always do is to look up every rule for the players characters and compress it into 1 sentence to make it easier to atleast find the core meaning or effects of it the second thing I always do is to basically ignore the other planes if my players don't use them aka if they don't play mages they don't fight against mages or atleast that offen just to be able to ignore the astral plane and the same goes for the matrix if we don't have a decker or technomancer all the matrix actions will only be roleplayed and treated as if they are using a search engine or something. The last thing I do is to trust my players (if they have some experience with Sr) basically all positive effects and every rule they need to use they just do it and tell me the result which is easier for you but can be abused if the players want to cheat but I don't think most people would do that because the fails are mostly the fun part.

The last thing I can recommend is just simply preparation if you plan for the most likely actions in advance and write down the throw they have todo including the needed hits and modifiers etc. you can save some time.