I want to buy an app, but don't want to waste money on something that's not gonna work for me. Is there any way to test drive one?
Sometimes app developers will release a version that allows you to play with it, but to get full functionality you need an in-app purchase. See this app for Summoner Wars as an example: https://itunes.apple.com/us/app/summoner-wars/id493752948?mt=8
But this is a choice the developer makes, so if the app you're interested in doesn't supply something like this (or a lite version; see @Alex's answer), then no.
If the app developer doesn't have a free/lite version, then no.
There is no facility for downloading a paid app as a demo or free unless the developer has an alternate delivery means. Some have review copies and/or a free version, but the store just isn't engineered to test drive an iOS app.
You can get someone who has purchased the app to install it on your device and in extreme cases - if you buy an app that doesn't at all match the description, you could ask Apple for a refund and document why their description doesn't match your experience. Although the store policy is no refunds - in practice, they are known to refund a purchase entirely when the reviewer didn't catch a problem and the goods were not "as described" when you purchased an app.
In this case, the shopkeeper is doing right by people that spent money based on their "review and stocking" process and got burned in an observable manner. You won't want to depend on this and instead be sure you are OK with spending money on something that may work as described but not meet your needs.
You can always ask the author beforehand about the specific functions you are curious about.
Ask beforehand if they are willing to provide a refund if the app does not meet your needs.
There is also the illegal alternative which is to jailbreak your device and download app files from a pirate/torrent site. You can test the app without paying, then buy it if it meets your needs. There are many ethical holes with this approach, but it does allow for testing without paying upfront.