How to sense current with OPAMP?
The circuit you show will work, however there are limitations. 

First, the output voltage cannot exceed the load voltage, in fact it can't quite get there because of Vce(sat). 

Secondly, the op-amp cannot swing very close to the supply voltage, so the output voltage cannot exceed about 2.5V with a 5.2V supply, 2 Volts would be safer. 

Thirdly, the input common-mode range of the op-amp is Vcc - 2V over range, so V2 must be no more than 5.2 - 2V = 3.2V. You need a higher voltage than Vbat to supply the op-amp. This is one reason why people are suggesting you buy the boutique parts- they are designed to have a common-mode range that includes the positive supply rail. This is probably your biggest hurdle- even if you're happy with 0-1V or 0-2V out, you need a higher supply voltage if your V2 is actually the same as the 5.2V, maybe 7.5V. 

Fourth, the offset voltage of that particular 2-cent op-amp can be as much as 9mV, which would represent an error of +/-9mA in the measured current. That can be zero'd out but it will typically drift +/-7uA/°C so for a 50°C change you could see 0.35mA change in the measurement. You can cancel out the error due to input bias current by adding 1K in series with the inverting input, but it's likely not worth it. Note that if the offset voltage of the particular op-amp you're using happens to be the wrong polarity you'll get no reading at all until the current exceeds the offset voltage divided by the value of the sense resistor since this circuit can only output positive voltage. So if your current to be measured is less than 9mA you may never see a signal (usually the offset voltage is better than that, but you can't count on it). 

And finally, there will be a small error due the finite gain of the transistor, likely less than the tolerance of the 1 ohm resistor, but probably more temperature sensitive.