Not much of a solution can be offered for this one but I can list some tips and my level times and hopefully it helps =)
1- Even if you have completed hard before this run the special technique is still required on the worlds 1,2 and 3 bosses.
2- You can restart a level if you feel it took too long however if you die and come back at a checkpoint the timer does not return to when the checkpoint was hit.
3- If you feel comfortable that you have enough lives to spare a couple and think you could rush a tricky part and reach a checkpoint try it and even if you die after the checkpoint you will be returned and will have saved time
For anyone looking for estimated times for each level here is a list of my level times including the total time at the end of each world and the end of the run.
1-1 = 3:51
1-2 = 5:01
1-3 = 4:15
1-4 = 1:07
Total time by the end of World 1 = 14:14
2-1 = 2:56
2-2 = 4:11
2-3 = 4:28
2-4 = 1:27
Total time by the end of World 2 = 27:16
3-1 = 6:34
3-2 = 6:19
3-3 = 2:21
Total time by the end of World 3 = 42:30
4-1 = 4:11
4-2 = 7:18
4-3 = 2:54
Total time = 56:53