This answer is simple and should be obvious. There are two methods of which one uses custom formats and another more easily editable solution.
1)
string Month = System.DateTime.Now.ToString("MMM");
2) Create a string array with all the names of the months and then simply index into it with System.DateTime.Now.Month to get month string you wanted. You might want to check whether January comes out as 0 or 1 and if the latter subtract 1 before indexing into the array.