I got lucky with a few good tie-ups from the gacha letting me get up in rank pretty easily. That's really the only way you're going to get into the 1-20 daily rank easily in any event with increased drop rate tie-up gacha items.
As for the 111,333, ect. ranks, as I said before, they are quite tricky to obtain. Your best bet trying for one of them is to try and get close to the rank during the day, and seeing if you can get slightly above it in the last 10 minutes (05:00 AM JST). If you get 112-115 or so, one or two people getting a higher rank will lower you into the winning one.
I don't recommend ever using energy drinks with an event with increased drop rate tie-up gacha items, as you can use a bunch of them and still hit a wall. If you do decide to use them however, try to do it on a slower day, like a weekday. Less people active, more chance to get higher.
In the end, a lot of it really comes down to luck. Luck on getting a good tie-up and luck on getting good present boxes. This event especially, as the wonderful present can give you severally hundred at a time. A few of them in a row can easily boost you to the top ranks.