I would like to purchase a basic SSL certificate, which allows me to secure a single domain (as opposed to a wildcard certificate for multiple subdomains). I was going to apply this to mail.example.com, but would I be able to simply use example.com and have the website and email use the same certificate? Do I simply change my DNS MX record from @ mail.example.com to @ example.com
you are right.
1.buy a normal ssl certificate for 'yourdomain.com'
2.create a mx record without mail. suffix
yourdomain.com. 800 IN MX 10 yourdomain.com.
MX domain on the right should be equal to ssl certificate domain. That's what program checks when sending emails. It helps your mails not ending in Junk mail.
3.The same ssl certificate can be used for a web server & mail server because a standard ssl certificate works for yourdomain.com & www.yourdomain.com (you don't need a wildcard ssl certificate) reference: https://www.clickssl.com/blog/do-i-need-different-ssl-certificates-for-www-non-www-domain
people will use yourdomain.com in their mail software settings to call the mail server. no need to point to mail.yourdomain.com
To answer your first question "would I be able to simply use example.com and have the website and email use the same certificate", No, not unless you buy a wildcard SSL certificate.
Wildcard SSL certificates secure your website URL and an unlimited number of its subdomains. For example, a single Wildcard certificate can secure www.example.com, blog.example.com, and mail.example.com.