The easiest way would be to open up the transmitter, desolder the potentiometer used for the stick and solder the cables to Arduino's PWM instead. Then use
analogWrite() to simulate stick position. If your TX would use that potentiometer with more than 5 V, you'll not able to use full ppm values. A RC or a RCL filter may be needed between PWM output and RC input to "smooth" the PWM into an analog signal. There are a lot of online calculator to find good R, C and L to polish your signal.
This assuming your TX is using potentiometer like stick; if it is a really cheap one, it may be using buttons. If this is the case, then you connect a transistor in parallel with the button, that way you can keep manual button.
Another way is to program your Arduino to look for ppm signal (normally 50 Hz with high starting value with a period from 500 to 2000 µs) and then look for this signal into the Tx PCB; if found cut the trace of the generator and solder your cable from Arduino. I've seen around a lot of libraries outputting ppm (they are used on drones to mix single ppm channel back to PPM to use only one pin from logic board without having to hack standard RX