Is it possible to create a passive income stream for under $1000? Where should I start?

Very possible, the hardest part is finding the area of interest.
What niche market or opportunity do you see yourself enjoying as a hobby. Enjoyment or purpose is key because passive income often requires putting in some time. At least at the beginning and on going.

Here is an example: I'm good at marketing, enjoy talking business tactics, sharing my expertise and like to write, although I'm not good at it. :)
One option I saw a few years ago was starting a strategy blog, this blog shares random thoughts, stories, tactics, tips, FYIs, etc. regarding business. I run Adsense on the blog to generat income from there. But my bread and butter when it comes to making passive income from the blog is the affiliate links I have on there. I enjoy reading and think I'm really good at finding great stories, bios, and How To books and I share some of those there, I think I have one of my favorite albums listed there and links to courses and services offered by other companies and some for my own companies (I run a web development firm, a business analytic and advertising team, and a hosting platform that is cheaper than GoDaddy) I have links there. All affiliates or ads to my other services.

All I do is drive traffic there once in a while and see the commissions add up. I do my homework in driving the right traffic.

Also, another example which is connected to the prior is that I have landing pages (see: trainer.unthink.me for example) where i get registrations for single page landing pages done by a contractor that works for me and we split the earnings. I don't do anything, but post them once In a while on Instagram.

I also have an affiliate system for online programs im vested in, and help drive traffic to increase the bottom line and get commission. I also don't do anything for this one. did take time and effort in finding the right market to advertise and message to use. Once that was done the rest takes care of itself.

Another example, a friend of mine has a blog about a particular lens piece for a camera loved by many professional photographers, they come to his blog (which he doesn't invest much time in anymore) to find information about tips and how tos for the lens... During the time he updated the blog daily and then weekly he would share his own affiliate links to Amazon and collects money that way. Because the niche was so targeted and there are a ton of people looking for that information he gets good return on that.
This is what you call a lifestyle business, but what the people that make money off selling lifestyle businesses don't tell you is that is for those who don't require much income.
Another passive income is investing, consider investing in family or friend that maybe wants to start selling Mary Kay or something, you invest in them by buying the products for them. The person sells you collect either an interest or perpetuating until you find some agreement of full repayment to you.

Buy a soda or vending machine if you have a truck and are handy with fixing stuff. You can always find deals on snacks and sodas and in a good spot a machine can generate average $50 per cycle. May not sound like much, but if you get yourself a couple of them you have a small income stream that can be saved and used to buy a candy machine or another vending machine... And give you weekend spending money.

These are just some ideas I hope help you get your brain flowing to see that you might not need money to do something and if you do you might have more choices than you realized before.