You can copy and transpose more than one column at a time. Highlight 10 columns, copy, then paste special transpose and all 10 columns will be pasted as rows.
Or, in sheet2, assuming your data starts in cell A1 on sheet1, enter the formula "=INDEX(Sheet1!$A:$Z,COLUMN(),ROW())" without the quotes, then you can drag the formula down and across as you suggested. Check out the INDEX formula in excel's help to see what it does. This assumes that your origianl data is in columns A-Z of sheet1 (hence the $A:$Z in the formula which you can change if necessary)