Any way to get the player's mouse position?

You can use the Mouse's .Move signal to run code respectively when the Cursor's position changes. Using Mouse.Y and Mouse.X in conjunction, we can get the current UDim2 position on screen, and assign it to our customary Cursor.

To actively hide the main Cursor, we can use UserInputService's MouseIconEnabled property to make it invisible

RemoveMouseLocalScript

local UserInputService = game:GetService("UserInputService")
UserInputService.MouseIconEnabled = false

CustomCursorLocalScript

local Player = game:GetService("Players").LocalPlayer
local Mouse = Player:GetMouse()

local Cursor = script.Parent

Mouse.Move:Connect(function()
	local MousePos = UDim2.new(0, Mouse.X, 0, Mouse.Y)
	Cursor.Position = MousePos
end)

Note:

Ensure you set the AnchorPoint of the customary Cursor GUI to .5,.5, so we can center it.