Question

def func(arg: String => Int): Unit = {
    // body of function
}

I mean this fragment:
String => Int

Answer 1

As mentioned in Advantages of Scala’s Type System, it is a Functional type.

The article Scala for Java Refugees Part 6: Getting Over Java describes this syntax in its section “Higher-Order Functions”.

def itrate(array:Array[String], fun:(String)=>Unit) = {
  for (i <- 0 to (array.length - 1)) {    // anti-idiom array iteration
    fun(array(i))
  }
}

val a = Array("Daniel", "Chris", "Joseph", "Renee")
iterate(a, (s:String) => println(s))

See? The syntax is so natural you almost miss it.
Starting at the top, we look at the type of the fun parameter and we see the (type1, …)=>returnType syntax which indicates a functional type.
In this case, fun will be a functional which takes a single parameter of type String and returns Unit (effectively void, so anything at all).
Two lines down in the function, we see the syntax for actually invoking the functional. fun is treated just as if it were a method available within the scope, the call syntax is identical.
Veterans of the C/C++ dark-ages will recognize this syntax as being reminiscent of how function pointers were handled back-in-the-day.
The difference is, no memory leaks to worry about, and no over-verbosity introduced by too many star symbols.

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In your case: def func(arg: String => Int): Unit, arg would be a function taking a String and returning an Int.