The problem is each line (except the last) includes a space. Although it may be invisible it affects the aligning.

So the solution is to move the whole text block by the width of one space (or similarly increase the textblock width by one space). Of course this moves the last line (which doesn't end with a space) a bit to far, so to equalize all lines, add a space at the end of the String. See the below example.

Adapted Code

1. String forecast = "Mostly cloudy in the morning, then partly cloudy. High of 63F. Winds from the ENE at 5 to 15 mph.";
2. 
   
3. size(500,750);
4. background(0);
5. fill(255);
6. textFont(createFont("Kievit-Regular",30));
7. text(forecast, 50, 20, 400, 200);
8. // >>>>>>> DEBUG
9. stroke(color(0, 255, 255));
10. noFill();
11. rect(50, 20, 400, 200);
12. noStroke();
13. // -------------
14. fill(255);
15. textAlign(RIGHT);
16. text(forecast, 50, 270, 400, 200);
17. // >>>>>>> DEBUG
18. stroke(color(0, 255, 255));
19. noFill();
20. rect(50, 270, 400, 200);
21. noStroke();
22. // -------------
23. fill(255);
24. textAlign(RIGHT);
25. forecast += " "; // add a space at the end of the String
26. float spaceWidth = textWidth(" "); // get the width of one space
27. // correct all lines for the end space by adding spaceWidth to x (aka moving the whole text block)
    
28. text(forecast, 50+spaceWidth, 520, 400, 200);
    
29. // >>>>>>> DEBUG
30. stroke(color(0, 255, 255));
31. noFill();
32. rect(50, 520, 400, 200);
33. noStroke();