The problem is the current is too low (about 0 - 4mA) and arduino gives reads zero voltage on the shunt.
If the load resistance is known you can measure the voltage across the load and calculate current, so you don't need a separate shunt.
Switched-fixed resistor values would probably be better than a pot because the values are known (without measuring) and it's easier to automate.
Measuring "short circuit" current might be tricky because you do need some resistance to measure the voltage. But, with the 1.1V reference you can get resolution down to 1mV and that should be "close enough" to short-circuit current for any practical purposes because you're probably not going to use the solar cell near zero-volts.