Our flight arrives at 7:20 pm. When I search with a pick up time of 7pm it quotes me a rate of $390. But if I change the pick up to noon (so essentially reserving the car for a longer period), it quotes me a rate of $350. Why would it be $40 cheaper for less time? And if I did go ahead and schedule the pick up at noon, but didn't show up for many hours (since my plane lands at 7:20) would there be any type of penalty?
Yep, Ralph&Pam have it correct, if you book it for noon but don't show up to pick it up until 7:30-8:00, your reservation will have already been cancelled.
The likelyhood is that you will be forced into a walk-up rate that will be much higher than anything you could have booked in advance.
You didn't say how long you will have the car and what time you plan to return it. It could be that extending the rental (by booking from noon) is pushing you into a weekly rate OR your expected return time could be affecting the number of 24 hour days that you have the rental.
Either way, I would just book the correct pickup time, which should be after your scheduled flight arrival time (say 7:30-8:00). Add your flight info to your reservation, in case there are any flight delays. And make sure your drop off time is reasonably accurate also (1-2 hours before your flight time)
Then check back frequently to see if the price drops. If price goes down, book a new reservation and cancel the original. Rinse and repeat.
I usually recommend a booking engine like carrentalsavers.com to check pricing as they apply any publicly usable coupons and codes to get the best pricing.
You can book through their site or just glean the codes/coupons and book directly on the car rental agency's website.
EDITED to note that sometimes trying your reservation using a different browser (IE, Edge, Chrome, Firefox, Safari etc) can, and often will, yield different results. Also try on a mobile browser (smartphone or tablet)