Remember the saying about two rabbits: the man who chases many things, finishes neither of them.
When answering to requests is only a part of your job duties, it is vital to maintain a schedule and procedures to distribute tasks.
I would recommend those having a crazy schedule: allocate blocks of time in the morning or later in the afternoon, when you have a fresh mind, to work on the tasks with a high priority.
Incoming requests may form sort of a black hole, that sucks you so fast that you will barely notice it. It is important to choose the right work rhythm and distribute energy during the workday in order not to lose the quality and be attentive to the needs of the customers.
Your goal is to find the golden mean: on the one hand, your basic work has to be done, and you need time to do this; on the other hand, you are expected to answer the customers' questions quickly. This skill is tough to learn, especially if you are not a part of the support team, but you can learn it from the colleagues. Play with your schedule. Do not try to answer all requests at once but divide the stream into groups of 5-10 tickets.
You can reach the balance by trying and falling down.