Absolute encoder vs Accelerometer for calculating angles
I'm hoping to get the X and Y coordinate accuracy within +/- 0.5mm. Is this accuracy feasible?


I'm only going to consider the A1 in your diagram since the accuracy of that will have the greatest impact in the final position. For the other angles any errors will have a smaller impact on the final position accuracy.

Take the most simple case, the arm is in a straight line almost completely in the X direction. 

You want a Y reading of +/- 0.5 mm with an arm length of 1400 mm. That will require an angular accuracy of +/- 0.02 degrees.

Assuming you have a maximum travel of 90 degrees and you scale things such that your full ADC range is used linearly over that range then you require log(90/0.02)/log(2) = 12.1 bits resolution on your readings. So after allowing for the noise in all sensor outputs you're looking at a minimum of a 14 bit data required to give that level of resolution. 

It's been suggested you use a pot connected to the internal 10 bit ADC. 
Assuming you used your full ADC range over 90 degrees and the 10 bit ADC reading had no noise on it you would be looking at around 2 mm accuracy. In the real world I'd be impressed if such a system could reach 5 mm accuracy.

Assuming you are working vertically accelerators could work, they typically have at least 16 bit resolution on the output but are noisy, you'd need to average for a long time in a very stable environment to get the levels of accuracy needed. Probably a non-starter.

A very quick google didn't find any easily available encoders with the required accuracy. I'm sure they exist but they are probably going to be expensive. You could possibly use gearing to improve the sensitivity of a more standard part but that's adding mechanical complexity.

In other words the accuracy you're aiming for isn't easy to achieve. You may want to look at relaxing your requirements a little.