I have a tablet/phone with a micro-USB connector. Is it possible to connect a USB drive to this port with an adapter and access the contents?
In some cases, yes. Whether or not you can do this comes down to whether or not a number of conditions are satisfied.
The connector: Most tablets and phones use a micro-USB connector, which flash drives can't connect to as-is since they use full-size USB rather than micro-USB connectors. That said, micro-USB-to-USB adapters are not hard to come by, so one of those can be used to attach such a drive to such a device.
The external power situation: Here's where things get especially problematic. Some external USB drives require power, either supplied by the USB bus or via an external adapter, to work properly. Some tablets, and pretty much all phones, don't supply enough power through USB for such a drive to work. For something like that, you'd need to power the device via another source to make it work. (And then you have to worry about the drive's formatting too.)
The operating system's support for external drives: Android and Windows should not pose a problem here. On the other hand, iOS does not expose the file system of the device to the end user. To that end, if a user wanted to plug in a USB device and explore that, he'd need an app specifically to do it. The iFlashDrive, for instance, is a combination flash drive and software suite for iOS devices that deals with both ends of the problem.