You've got the right approach. One slight problem with your equations: 7^732 == 7^400 * 7^332 == 1 * 7^332 == 7^332 (mod 1000) (you had addition instead of multiplication).

It's easy enough to calculate from here. Since 332 = 256+64+8+4, just calculate those powers by repeated squaring:

7^2 = 49

7^4 = 49^2 = 2401 == 401

7^8 == 401^2 = 160801 == 801

7^16 == 801^2 = 641601 == 601

etc.

It's easy enough to drive to the finish from here, which would be a reasonable approach.

You might notice the "01" pattern and figure out that 7^20 = 7^16 * 7^4 == 601 * 401 = 1 (mod 1000). Then you would calculate 7^332 = 7^320 * 7^12 = (7^20)^16 * 7^12 == 1^16 * 7^12 = 7^12 == 201 (mod 1000).

Since you know 7^400 == 1, you might try dividing out the prime factors of 400 and calculate 7^200 == 1, then 7^100 == 1, then 7^50 is not 1, so going back to 100 and try dividing out 5, so 7^20 == 1, and then 7^4 is not 1. That's a lot more work than the original problem, though.

Post again if you still have trouble.