Selfhosted

You don't

First off one person does not equal interest. Unless there is a decent amount of people wanting this it isn't worth there time.

Second and most importantly there is a serious issue with moderation. Who is going to do the policing and who gets the final call? When drama goes down there will be lots of blame and anger. Ultimately everyone will be pissed off.

In short I think this is a bad idea. What I would do instead is start a community fund to run a mastodon server in the cloud. Be the change you want to see. Just don't be surprised if no one wants to get on board.

“How do I convince my tech department to take on additional tech debt”

You can always ask the student body. If they're doing a good job, they're networked and know people and procedures. Sometimes the IT helpdesk people are knowledgeable and know who makes those kinds of decisions.

And I think server hosting and paying for that might work differently than in normal life. A university has quite some IT infrastructure. Maybe they have a free VPS to spare for things like that. Maybe it has to be super secure, intergrated into the single sign-on... It's more a political decision. Could be anywhere from free, to you need to pay half a person's salary to moderate and maintain the instance to their (high) standards.

Unless they have an active IT department that is already hosting various websites for them, it is probably better to show them some hosted offer like this: https://masto.host/

As for your questions:

1. Medium difficulty as far as self-hosting goes. Mastodon is a Ruby on Rails app.
2. Really depends on the number of users etc.
3. See no. 2. a few hundred $ or so.
4. As others have mentioned, let them make an account on an existing server first, it can be migrated to your own server later.

Why not start with an account? You do not need to run a server to be int the fediverse. It's enough for now if they've got a or multiple accounts and actively use it. Share the latest papers via that account. Provide news. Link to articles etc. That's good enough for the beginning.

Honorable goal. But it won't work. From a business's perspective, hosting such a server adds inherent risk. They won't do it. You may be able to convince them to use the fediverse through threads or bluesky, but they won't host their own instance, even if the CTO/CIO/CISO agrees with your love for the verse. I'm not saying you shouldn't try. I'm saying you should provide alternatives to twitter and to self-hosted mastodon; alternatives that don't require self hosting anything.

They absolutely won't have better reach. Mastodon is significantly smaller than both Twitter and Bluesky.

"that's the great part -- you don't!"

But Honestly, it'd be pretty damn cool. I wouldn't be too surprised to find they don't even run their own mail server though.

Might look into Gotosocial considering it's more lightweight.

Other than that I wonder whether it's better for the uni/department to register on another instance, similar to how fosstodon is home for many FOSS projects. An inter-unis, academic-focused Fediverse server might be a good club project, and could allow more donation sources to sustain too (although this is pretty above-scope from what you're asking).

It’s not really any different than hosting any other service.