Yes; I do everything solo, for the most part. They're generally pretty easy.

If you have a save you consider your "main" I recommend using that character to start the expedition at the nexus.

Its not a "timed event," per se, the expedition will be active for six weeks. You are able to return to the nexus and go back to your main save at any point and continue the expedition again in the future picking up where you left off. So if you have other stuff you need to do like harvest plants at your base you can go to the nexus, go back to the main save to grab your stuff, and then go back into the expedition when you’re ready without any hitch. You can also bring a limited number of items with you to the expedition if you start in the nexus, and you can bring stuff back to your main save when you finish the same way. Thats why I recommend starting on a save you already have.

You'll start basically a new save with your character and follow the story to unlock rewards. You respawn on a random planet and have to go find your ship to repair it and start the expedition proper, then once you're in the air you can start doing the expedition goals to get power ups for this "new" character and unlock expedition awards as you complete all the goals in each tier. (You can return to the nexus after doing your first warp jump to get your stored resources and tech)

Usually its simple stuff like harvest a bunch of plants or scan some number of plants and animals and then as you go further along they get more specialized around the specific expedition.

You and your friends can play in your own saves and join each other at any point to complete objectives in tandem without impeding your private progress or requiring you to both be on the same step. And as long as you can play NMS you have access to all updates and dlc.

Hopefully this answers most of your questions and gives you a platform to dive in! Imo this is one of the most impressive updates I've seen outside of worlds 2. You definitely want to get in on this!

Expeditions run for anywhere from 4-8 weeks, but they usually only take a dozen hours or less to actually do. I've been able to dedicate a few hours on a weekend and do the whole thing in a session or two. If you're familiar with the game's systems, it makes things easier, since you already know what to do to accomplish most of the goals. Plus, you can now convert an existing save to a expedition save and bring some of your equipment and supplies over. Once you're done, you can convert back and bring your stuff back, along with any new things you found. They've made it pretty easy to engage with.