$$Rate\ of\ printing\ =\ \frac{total\ number\ of\ paper\ printed}{time\ taken\ \left(in\ hours\right)}$$
Question=> how many news paper did the printing press print on sunday morning?
Total number of paper printed = rate of printing * time taken
Time taken = 1:00am to 4:00am = 3 hours
Rate of printing = ?? unknown

Statement 1 => The rate of printing on Saturday morning was twice that of Sunday morning
Rate on Saturday = rate on Sunday * 2
Rate on Sunday = $$\frac{Rate\ on\ saturday}{2}$$
But the exact value for the rate on Saturday was not given so we cannot find the rate on Sunday. Hence statement 1 is NOT SUFFICIENT.

Statement 2 =>On Saturday morning, the printing press rant at a constant rate from 1:00 am to 3:00 am, stopped for a half-hour and then ran at the same constant rate from 3:30 am to 5:30 am printing a total of 4000 newspapers
Total newspaper = 4000; time taken = 2 + 2 = 4 hours
Printing rate on Saturday = $$\frac{4000}{4}=1000\ papers\ per\ hour$$
This statement only refers to Saturday. Hence, statement 2 is NOT SUFFICIENT

Combining both statements together =>
Rate on Sunday = $$\frac{Rate\ on\ saturday}{2}=\frac{1000}{2}=500$$
Total number of papers printed on sunday = 500 * 3

Both statements together are SUFFICIENT
Therefore, answer = option C