In order to even unlock this mode, you must have completed the game. I would not attempt the Expert Course until you have obtained all 10 Champion Medals and are pretty confident in your abilities to run through world's 8, 9, and 10 without using a continue.
The Expert Time Attack Course requires that you play through all 10 worlds without using a continue, meaning a complete attempt is going to take you anywhere from 90 to 120 minutes. This is tricky, because you won't even get to the difficult levels for at least an hour, meaning that you're investing all that time just to have an attempt at the achievement.
During my Champion Medal run throughs, I had the most trouble with World 10 Stage 5, but was able to build a solid strategy with YanYan until I was having more regular success. Here is the clip during my attempt where I got past this level.
I was extremely lucky and got through the entire Expert Course on my first attempt. I had built up 45-50 lives through the first six worlds, and then proceeded to lose half of them with some crazy complications on World 7. By the time I got to World 10, I only had 10 lives and thought I was toast. Luckily, I made it to Stage 5 without much fuss, and got through it on the first try (video above), and the following stages weren't too bad.