A Daily Challenge badge is rewarded at the end of each month,
so this one will take you a full ten months to complete. 
EDIT: If you have premium, you can go back to September 2020 without touching your phone settings, making the achievement available as of June 2021.
If you don't want to spend money on premium, then you can go into your phone's settings and set the date to 31st of Dec, 2020, and you will have access to November and December, saving 2 months of your time!
More on this in the "Note on Date & Time + Warning:" spoiler.
To earn them you need to do Daily Challenges and earn points from them, luckily, each badge includes the ones below it, so if you earn Gold, you will also get Silver and Bronze.
- Bronze ≈ 30 puzzles, or about six days of puzzles.
- Silver ≈ 60 puzzles, or about twelve days of puzzles.
- Gold ≈ 90 puzzles, or about eighteen days of puzzles.
- Diamond ≈ about 120 puzzles, twenty-four days of puzzles.
There's a total of 140-155 puzzles per month, so you've got some room if you find some too difficult.
Note for people reading in/after Oct 2021:
*** Spoiler - click to reveal ***You can change "month" in the top right corner of the Challenges tab. You can go back and do the past months. You can also change year with the arrows at the top of the calendar. However, the game released to mobile in January 2021, so it won't be until mid-late October 2021 that this achievement will be available.
Note on Badges:
*** Spoiler - click to reveal ***Each challenge is a different difficulty, and reward different levels of coins. At the time of writing, both the number of coins required for each Badge and the number of coins rewarded, are not shown. The above is only a rough estimate of how many puzzles you will need to solve for each badge, and will vary depending on how many of each puzzle you do. It will be fastest if you only do hard puzzles, but they will also take you longer to do.
Note on Date & Time + Warning:
*** Spoiler - click to reveal ***As JettSeriesX has said in the comments, you can go into the settings of your device and change your date & time to the end of last year, which will give you access to the puzzles from the 20th of September, 2020.
I would be cautious about doing this, as tracking on Microsoft's mobile games can be picky. I would do this at the start of a month before you've done any challenges for the current month, or at the end when you've done all of them that you need to do. This should give the game enough time to update. Noodles Jr said it took his game a few days to update before it showed the progress, so just be patient!