I was a college senior in 1985, and my dad was a tech hound, chasing the new everything. From memory, mostly, so I could have some things wrong:
You'll want to be more specific about the brand. The Amiga came out in mid-85, but otherwise your guy probably had a Commodore 64. Worth keeping in mind that the IBM PC came out in 81, and (according to Wikipedia) was eating everybody's lunch by 85. (My dad went straight from a PET to an IBM PC.) Commodore stuck around for a while, but the writing was on the wall.
That said, email always required logging in to something else. It also wasn't a particularly common way of communicating, outside of universities and a few tech companies. Communications would almost certainly have been phone or snail mail, or telex, fax, or even telegram. Voicemail wasn't in widespread use; your character would more likely have had a tape-based answering machine that had the potential of filling up.
The term "internet" wasn't widely used (IIRC that started around the time Mosaic came out in 1993). People chatted over Compuserve, or they would telnet into a server and connect to Usenet. Compuserve forums were proprietary; Usenet was public if you could get to it.
The TL;DR here is that in 1985 most people were aware of computers, but they were not considered indispensable household tools (most of the families I knew didn't have one). Email existed, but it wasn't considered a primary communication source. If your character is an ordinary person communicating in ordinary ways, they would probably need a physical telephone somewhere, and a way to store messages. Snail mail can be sent general delivery, but if they're receiving faxes or telexes, they'd probably have to pay a service.
If this is a tech thriller, of course, your options are wider.
But there would have been less for this person to hack into as well.