What happens to emails that attempt to be delivered to my box while MiaB is being rebooting, or, when the the Digital Ocean disk is being resized (which of course is a much longer operation)?
The sending SMTP server will retry at time intervals up to 48 hours or longer until the email is delivered or the maximum time is reached.
See RFC 2821
4.5.4.1 Sending Strategy
https://www.ietf.org/rfc/rfc2821.txt
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What @paradoxbound said is true, but I would like to add that if you are concerned about mail server downtime, you can setup another mail server (you should probably pick a server data center that is physically located somewhere else), I believe you can add a second, third, fourth, … mail server record (MX record) as failover servers with the same hostname, but increment the priority (1, 2, 3, …).
EDIT: This is how it looks on Namecheap, my personal favorite DNS/Registrar people for domains.
I would change those MX priority numbers to 10, 20, 30 etc. That way you can insert new records between them on short notice.
See Googles MX record:
$ dig +short google.com MX
10 aspmx.l.google.com.
30 alt2.aspmx.l.google.com.
50 alt4.aspmx.l.google.com.
20 alt1.aspmx.l.google.com.
40 alt3.aspmx.l.google.com.