Its going to be tough finding a fully managed VPS in that price range. You can find plenty of un-managed VPS, but as you have probably found, there's a lot of work involved in installing software, securing the server, and maintaining it day to day.
If its not a "mission critical" application where you livelihood depends on the server, there are some options. You could go un-managed and expect to take the time to learn the ropes, or hire someone on a one-time basis to set up your un-managed VPS with a free control panel like Kloxo, a firewall, and whatever other software you need for your site. Expect to pay $65 to $85 per hour for this kind of work (and probably just an hour or two).
The other option is to find a host that offers something in between managed and un-managed, where they set up and do basic updates of your server, but don't offer the same 24/7, rapid response support that you get with full management. They are out there, but per forum rules we cannot recommend ourselves or any host we haven't actually used.
This budget is too low for managed solutions. Mostly, managed VPS come with cPanel which is about 15$ itself. Try to raise up a bit your budget or prepare to suffer from bad quality and oversold resources.
Maxence H. - UBservers.com
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Great! Let us know about your experience with them after a month or so. Because of forum rules, we can't recommend a service we haven't actually used. That leads to frustration from people who pop in and ask for recommendations, but its unavoidable given the rules.
But if you leave a review, we can at least point someone else looking for the same thing to your post.