Microsoft Project is the first thing that springs to mind that could help with the practicality - but you've reminded me of my first day on a previous job, which sounds a bit like yours.
It was a small company, and I was hired to be the only hardware engineer. No problem, in theory. The company's products were within my area of expertise, and I was more than capable (technically) of doing the job. But that wasn't the issue...
On the first day of the job I sat down with my new line manager in his office, and I asked him what I'd be working on. I expected to be busy, because after all, that's why companies hire new staff. There's no point taking on a new engineer if you don't have plenty for them to do - right?
He proceeded to describe to me a project that needed working on. And another, and then another, and so it went on. I counted about 10 in all. Cool, I thought. Plenty of new jobs to get stuck into.
I asked which was the one that was most important, and needed my attention first.
"All of them" was his reply. And he wasn't kidding. Every individual job was the single most important thing I could possibly be doing.
An analogy: Imagine writing some code in which every process and every interrupt has a priority of 1, and you'll get the idea. Make sure to run it on a single core - me, in this case - and see how much time there is for sleep().
It's hard to get much done when most of your bandwidth is consumed by task-switching overhead. Silly me thought that the job of a manager was to, well, manage resources. How naive I was back then.
Now I run my own business I still have as many different jobs to do, and of course, they're all the single most important thing I could possibly be doing. But now I'm the manager.