I was part of a small company, and sometimes we had to bring down our Exchange server to perform maintenance. The majority of mail during the downtime would get redelivered in a timely manner, but sometimes important emails would take days to finally get redelivered.
Is there a way for another server to accept the incoming email and then pass it along after the maintenance window is over?
Absolutely. You're looking for a secondary MX (mail exchange) service. There are a variety of companies that and provide such a service. I like DynDNS.com's "Backup MX" service myself, but there are many companies who can provide such a service.
You could provide such a service yourself with a secondary mailserver on-site, but if you don't have redundant Internet connectivity to your office it probably makes sense to get this service from a third-party anyway (so that your email doesn't "go nowhere" if your Internet connection fails).
What you can do is either provide a backup MX record pointing to some front end smtp server (I used to do that with a linux box with exim) which is configured to redeliver your mail to the exchange server, or -why not- have that box in front permanently (as your public smtp server). The lightest servers will do really, so that solution should be in reach of any kind of organization.
Having a backup MX record and server is the way to handle this one. Other comments on this topic cover that well. One thing we've run into is that you have to have the same quality of spam-checker in front of your backup MX server, as spammers frequently specifically use backup MX servers under the assumption that their anti-spam is not as effective.
This is one of the reasons we "outsourced" our email virus scanning + spam filtering to one of those offsite providers. Not only does this remove what can be an intensive process somewhere else, and save your internet connection bandwidth by not having stuff that you're going to reject out of hand travel over your link, it also means that in the event of an outage any email is queued by them until your server can pick it up. Obviously this costs a bit but depending on how much email you get and how important email is to your business, it might be worth considering.
We used Messagelabs and have since moved to Mimecast, but really there are dozens and dozens of providers out there you could look at.
As has been said, the easy part of the answer is a secondary MX pointer. The hard parts are generally how many (it depends) and where to host any secondary SMTP servers.
If you are located in an area which is prone to Internet outages or your hosting provider is prone to outages (still happens), then arranging for secondary SMTP hosting at a stable location is "A good thing to do"tm.
If you only have one Internet link, then moving up your provider chain to the most stable provider (generally your service provider) would be the best action. This also allows for email store-and-forward when your site is offline due to issues beyond your control, (Nature, utilities, construction, etc.). Availability and cost vary by provider, but the service is generally a negotiable item.
If the secondary SMTP host is configured to accept all mail and only forward to you, then the issue of SPAM and Virus checking becomes a single point, but this is rarely the case and you would need to have this as part of your secondary agreement so your domain is not flagged as promoting due to inaction.
Multiple Internet providers allow a much more interesting set of solutions to this question.
We use Frontbridge (oops, sorry, Exchange Hosted Services) to do anti-spam and anti-virus filtering on our incoming mail, and it also queues mail when your server is down. Here's MS's information page:
http://www.microsoft.com/online/exchange-hosted-services.mspx
