You need to complete Daily Polarium for at least two "full" months (i.e. your first month would not count unless you start on the first day of any month). You will unlock either a tile or cursor set on the first day of the following month, depends on the month (see below). The last set will be unlocked when you finish a year in Daily Polarium.
Note that you don't need to complete the secondary objectives (e.g. start and end on specific point etc) to unlock these. Once unlocked, these can be chosen in the option menu.
These sets can be chosen in teh options menu, once unlocked.
Once unlocked, this mode is accessible from the Time Attack menu.
Insert your copy of Polarium Advance into the DS GBA Slot when playing Actionloop (Puzzloop) and you will unlock an extra Tile set on Polarium Advance.