World 4-2 is as challenging as riding whales. In map A, you might have some trouble with all of the Beezos flying at you. The best strategy is to make small jumps and keep moving. At map B, you'll have to walk across the backs of whales.
Balanced character selection
This is actually going to be the first stage in a three stage stretch for the Princess. You're going to become very familiar with her over the next few worlds. Her ability to really zero in on a landing spot, such as tiny whale tails, will be of tremendous use to you here.
Map A
Advance your lives: only in Super Mario Advance, the room you start in will contain a big POW and several Porcupos. Throw the big POW to knock out all the Porcupos and gain a 1-Up. Then climb up to the next area.
Map B
Part 1
(1) Get Flurry! Take care of these pesky Flurries now, or they'll keep coming after you. Even worse, you'll be attacked by them and Beezos at the same time.
Part 2
(2) Hop carefully. The Beezos that fly toward you fly in a very specific pattern. If you practice enough, you can get the timing down and jump over them successfully every time. It takes a few tries to get the rhythm down.
Map C
Part 1
Whales, Beezos, and Flurries
Oh my! Hopefully you got the hang of running on ice during the last stage. The trick is to not lose control of your speed—it's hard to stop once you get going. Whales are the new "enemy" on this stage although the only thing you need to watch out for are their spouts. You can land on top of them safely, but don't walk through them.
(3) Sub-space 1. Before you go running off to the right, it's worth taking your time to investigate the bit of stage to the left. Here you'll find grass growing out of a whale's back. Pull the left grass up to find a potion. Enter into Sub-space here to find the Mushroom on the whale's tail. This is where the Princess' jump control can come in handy.
(4) Sub-space 2. Although there's a Potion to be found among the rocks, try as you might, you won't find any Mushrooms in the vicinity. That means the best use of this Potion is to collect six coins that can be found in the grass nearby. However, it may have one other use...
Super Mario Advance Mushroom
In Super Mario Advance, on the other hand, don't be so quick to dismiss that second Sub-space potion, as it can now serve a more immediate purpose. You won't even have to go too far with it. Just drop it down near where you collected it and step through the door. For your troubles, you'll find another hidden mushroom.
Part 2
(5) Rise across the water on the whale spout. The whale spout will carry you up to where the cherries are, but be extremely careful. If you blow it, you've got big problems. The Princess' hang time in the air will be of great assistance here.
A secret warp
Grab the Magic Potion from the rock in the first part of map C and go to the right, past five whales, to find a jar on a platform beneath some ice. Place the potion next to the Jar, enter Sub-space, and climb down into the Jar to warp to World 6-1.
Map D
(6) Autobomb and cherries! You can't walk on the floor—it's covered with spikes! Even the Princess can't float over the long row of spikes. Ride on top of an Autobomb instead. You may have to duck down if you're tall to stay on the Autobomb. If you collect the cherries, Starman will show up, but he may not appear in a particularly helpful location. Focus on reaching the other side. It's not a bad idea to bring a Potion along for the ride if you want to get stronger.
Map E
(7) Birdo on ice. Birdo is back, but the battle is on a slick surface. This Birdo has a random pattern of eggs and fireballs, but the fireballs are rare. Jump straight up in the air when her eggs approach, and don't run at Birdo. Wait for her to come to you before throwing an egg, or you'll slide into her and take damage.
After defeating Birdo, grab the Crystal Ball 
and head through the hawkmouth gate.