(I'm assuming US) a radar detector is definitely worth imo. It practically pays for itself if it prevents a speeding ticket at least once, but the main thing to keep in mind is that it will only pick up cops that are actively trying to detect speed. Meaning, you should still be on the look out for cops on the road and not to confuse it with a "cop detector". And a lot of times, it picks up other noise (toll booths, random security systems, etc) -- it takes a little while to get used to differentiating false positives.
But after a while, it becomes kind of like another sense available that you can use as you drive to know when/where you need to slow down. And if anything, it's peace of mind too. You'd honestly be surprised by how many cops you pass on the road that aren't actually tracking your speed (these by the way won't show up on the radar detector because the cop isn't actively radaring you). As long as you slow down and don't give them any reason, you should be okay.
As per cost, I'd budget around $500 to $700. I found this one to be quite good: https://store.valentine1.com/Store/item.asp?i=20020
Edit: (to answer the night/day question) time of day makes zero difference in a radar detector because science. It's the same level of effectiveness throughout the day. Just make sure it has a clear view to the front and back windows and you should be all set.
Poor assumption on my part. As for cheaper alternatives, I don't know any off hand but I'd imagine you'd have to do the tuning yourself (for better or for worse) which means your false positives will vary.