No, probably not. Your $5 VPS is going to be dedicated to you. The VPS provider is going to provide no privacy guarantees to you and will easily be able to associate requests for your information with your VPS. They will have no issue providing it to whomever asks for it.

In addition, your traffic will be the only thing coming from that VPS, so tying network information to you will be easier.

The key thing to remember about VPNs is that you are re-positioning your trust. Instead of trusting your ISP with your traffic, you are trusting the VPN provider. A hosting provider is about as trustworthy with your traffic as an ISP, and offers the same amount of pseudo-anonymity. It's a lateral move at best.

VPN means many things of which, you would be throwing away anonymity as you're completely stuck with a single IP forever which no one else will use.

Speaking from experience, definitely not.