Architecture

Before you YOLO deploy your Next.js app on a $3,99 VPS, it makes sense to take a few minutes and think about the overall architecture of your application.
Since you are not hosting on Vercel, you do loose all those nice benefits that come with it.

Hosting architecture for self hosting Next.js
Exemplary architecture of a self hosted Next.js application with a DNS/CDN layer infront of a VPS with Next.js, a proxy (e.g. Caddy) and an image optimizer (e.g. ipx) running as a Docker container

DNS & CDN

Maybe not relevant for your hobbiest side project, but for a "professional" project you should think about having some kind of DDos protection and/or a CDN caching your assets.
The commonly known providers (e.g. Cloudflare, Bunny, etc.) often do offer that in a bundle. With most providers a free tier is available.

Media Handling

Unless your whole applilcation contains just a hand full of images, you should probably not rely on the Next.js internal image optimization (based on sharp), to reduce the load on your server.
Using a custom image loader is the way to go here. an image and video optimization platform (check out the Next.js docs for recommended providers).

optional: Proxy

This really depends on your setup: if your are choosing the self hosted Docker route, you are going to need a proxy/app server routing the traffic to your Next.js application.
Popular choices are NGINX or Caddy. With Caddy being a little bit more end user friendly, since it does handle SSL/TLS termination for you. And NGINX being more performant, but a little heavier on the initial configuration.