Is there any short-way to find first '1' bit?

I want to show ,the mentioned problem on the title, to you in a function.

finding_first_one(signal a : std_logic_vector(...)) { return bit_number }

Meaning this, lets say, we have a signal '10010100', then return value, bit_number, should be 2. Is there any short way to find it in one cycle. I do not want to scan all the bits per clock.

You can do a for loop in your function.
Be aware that for loop cannot always be implemented on hardware and that it can use A LOT of logic elements.
Try something like this (not tested). index should be equal to 2 in one clock cycle.

architecture behav of test is

  signal sig    : std_logic_vector(15 downto 0) := x"2224";
  signal index  : integer;

  function finding_first_one (signal a : std_logic_vector()) return integer is
  begin    
    for i in a'low to a'high loop
      if a(i) = '1' then
        return i;
      end if;
    end loop;    
    -- all zero
    return -1;
  end function;

begin

  process (CLK)
  begin
    if rising_edge(clk) then
      index <= finding_first_one(sig);
    end if;
  end process;

end architecture;