The issue with a blue rush archetype is that most of the best cards in blue either have jump or require setup on an opponent's well. For the exact same land cost it takes to get to your opponent's face (5 lands), you can use those 5 lands to go to the sides instead and set up both a double collecting jumper and threaten your opponent's face anyways. Cards with enemy well setup take time (such as colossus requiring discounts over time or mystic beast requiring 3 special lands) - time which could be spent double collecting instead of awkwardly waiting at your opponent's face.
As a general rule, rush creatures should have either very low land counts or no lands at all (neutral). Blue has 2 very solid creatures that fit that description: Sharshell Keeper and Tyranax. What they lack in jump, they make up for with bulkier stats. This makes them the best rush creatures you can have in blue. Starshell is actually pretty nice here because it lets you draw, which is a resource rush is always lacking in.
Shifting Tide is also a very nice card in this because it allows you to drop a blue creature on turn 2 in the center of the board and still threaten face. You can also use it to shifting away any defenders or to get a nice juicy Daring Adventurer.
Compass is only in here because of the Starshells. I'm not sure it's really worth the 1 Faeria investment but you should be able to get 1 treasure pretty guaranteed out of it in a match.
Baeru is expensive but the early land destruction can win you certain matchups on the spot so it's definitely worth the include.
3 land cards are super expensive for rush lists so that's why there is just 1 Ninja Toad here as a late game tech, but mystic beast felt too good not to include. They may be worth cutting for something more neutral though.
If you don't have DLC's you could try using other neutral creatures like Maceman but I think Starshell really makes this deck. Without DLC's you start to run out of useable neutral creatures and there aren't any other blue creatures worth putting in here either.
You can find a sample match below - I have made changes since then: Bloated Toad was just not good enough (body blocking and Daring Adventurer synergy was my idea) but when I cut Bloated, that also made Aurora pretty weak so she had to go too. I also swapped Commands for Campfires because of the cost difference and you really don't need healing or structure damage in a rush deck.
Anyways, this is the best I could come up with for a blue rush archetype. Will it compare to yellow or red rush? Absolutely not, but it's a fun little idea to play around with