Sorting incoming email faster is one of the critical ingredients of email mastery. You need to be able to process without acting on things. First, know what you have. Second, do the right next thing.
Taking as few seconds per message as possible whip through your inbox and delete, file or label everything as appropriate.
If you don't already have labels set up for your mail, do that quickly now. These labels should reflect the order in which you need to deal with things and I find it helpful to group by how long I think it will take to move this issue forward or perhaps even resolve it. My labels, once again, are Urgent, 2 minutes, 10 minutes, 30 minutes and Waiting For Someone. Also I recommend creating a Tickler folder for things you don't need to deal with until a certain day.
Before you begin, decide generally what you file and what just gets deleted. The only things that remain in the inbox are those that require some action.
If you have a very full inbox, work your way through this in 10 minute bursts. Set a timer (I recommend Minuteur for Mac users) and do not let yourself get distracted in that time. It's just ten minutes; you can do this! If you aren't done after 10 minutes, then go deal with some other discrete task that needs to be handled today – preferably something on paper or involving talking to someone so you get a break from the intense mousework – and then when you've had the pleasure of checking that off, come back and do another 10 minute burst. Repeat as necessary.
For some messages, the obvious action will be to give it the 2 Minute label because it's a long message you know you have to read and may need to act on, but your few second scan of it makes clear that that could wait until tomorrow if it had to. Read it later, even if later is just after this burst. Do not act while you're processing the inbox – you are only spending a max of 10 seconds per item to sort it into stacks.
Those folks who regularly need to send the same reply will benefit from creating a few mail templates to make this a matter of a few clicks rather than a bunch of redundant typing. If you can say the usual thing without having to create it afresh every time, you can often close out those kind of messages in your processing – just be sure you're only doing it on the ones you can truly handle in 10 seconds or less. The ones that need the standard message plus one or two more sentences belong with your "2 Minute" label.
Your goal: a prioritized inbox which contains only things requiring action and all the contents of which are generally familiar to you.
This will reduce your stress, help you focus your time on the most important actions, and give you the ability to respond much better to demands for status reports.