Is it possible to upgrade a second generation iPod Touch to iOS 4.3?
According to the incredibly useful iOS support matrix v 2.0 (Winter 2012) done by Empirical Magic, the end of the line is running iOS 4.2.1 on an iPod touch (2nd generation). 



Apps that target 4.3 or newer are just not going to run on that device unless the developers re-add support for that OS. Since the code set for that device is ARMv6 you can expect more and more apps to be dropping support as time goes by. Thankfully, what you have should keep running indefinitely, but updates will taper off.