Meme even in the modern internet sense, was used in the 90's, here's Memepool from 1998:
So yes, we can be certain that "going viral" existed 20 years ago, to the extent that we think "viral" is the status achieved by "viral videos".
the modern sense of "meme"
> 2. An image, video, piece of text, etc., typically humorous in nature, that is copied and spread rapidly by internet users, often with slight variations. Also with modifying word, as internet meme, etc.
> 1998
> The next thing you know, his friends have forwarded it [sc. an animation of a dancing baby] on and it's become a net meme.
> Sci. & Technol. Week (transcript of CNN TV programme) (Nexis) 24 January
the modern sense of "viral"
> Chiefly Marketing. Of, designating, or involving the rapid spread of information (esp. about a product or service) amongst customers by word of mouth, e-mail, etc. to go viral: to propagate in such a manner; to (be) spread widely and rapidly.
> 1989
> The staff almost unanimously voted with their feet as long waiting lists developed for use of the Macintoshes... ‘It's viral marketing. You get one or two in and they spread throughout the company.’
> PC User (Nexis) 27 September 31
Edit: format