How would I stop this event from firing multiple times?

Use the MaxActivationDistance property of the ClickDetector and set it to 0, doing this will make it impossible to click the detector and activate the function. Then, after however long the animation takes you can set it back to the default value.

ClickDetector1.MouseClick:Connect(function()
    ClickDetector1.MaxActivationDistance = 0
    local Arrow2X = Arrow2.CFrame.X
    local OldArrow2X = Arrow2.CFrame.X
    local Arrow2TorsoY = Arrow2.CFrame.Y
    local Arrow2TorsoZ = Arrow2.CFrame.Z
    local Arrow2Ori = Arrow2.Orientation
    wait(4.25)
    if 2+2==4 then
        for i = 1,145,1 do
            Arrow2X = Arrow2X + 0.59
            Arrow2.CFrame = CFrame.new(Arrow2X,Arrow2TorsoY,Arrow2TorsoZ)
            Arrow2.Orientation =    Arrow2Ori 
            wait(0.005)
        end
        for i = 1,10 do 
            Arrow2.Transparency = Arrow2.Transparency + 0.1
            wait(0.1)
        end 

    end

end)