Yes, you will need an AVR ISP programmer to upload the new firmware, which is a *.HEX file.
BUT, things are a little more complicated than that because the latest TransistorTester revisions have extended functionality and require the mega328 AVR microcontroller, which has 32 kB of FLASH memory. Most (if not all) of the chinese TransistorTester knock-offs have the cheaper and smaller mega168 that has 16 kB of FLASH memory only or, even worse, the even cheaper mega88 with 8 kB of memory or the older mega8. The latest firmware will only fit in 32 kB memory, if you want it to have full functionality; or else, you will have to recompile the project with some functions disabled (read the project files, I do not remember specific details) in order for the firmware to fit in FLASH memory sizes of less than 32 kB.
So, with the ISP programmer, you will firstly need to check which microcontroller is on board (since most of these knock-offs usually have their chip identification marks erased). If the programmer identifies the microcontroller to be a mega328, you will just have to load it with the latest firmware and you are done. If the chip is identified as a mega168/mega88/mega8, you will also need to change it with a mega328.
-George