I haven’t benchmarked, but you could do something with a for comprehension

iex(103)> map = %{
...(103)>     int_a: "1",
...(103)>     int_b: "not_a_int!",
...(103)>     other: "leave me alone"
...(103)> }
%{int_a: "1", int_b: "not_a_int!", other: "leave me alone"}
iex(104)> for {k, v} <- map, match?("int" <> _, Atom.to_string(k)), into: map do
...(104)>   case Integer.parse(v) do
...(104)>     {int, _} -> {k, int}
...(104)>     _other -> {k, nil}
...(104)>   end
...(104)> end
%{int_a: 1, int_b: nil, other: "leave me alone"}

Thanks people for all the responses!

Indeed a lot of way to approach the problem, I will probably use the @amnu3387 response just because is a little bit faster, but as Eric said, is just a matter of taste, and all other options would be completely fine.

Thanks again for the help!

The benchmark results were:

Name                             ips        average  deviation         median         99th %
mnussbaumer_response        927.74 K        1.08 μs  ±2239.93%        0.96 μs        1.98 μs
kokolegorille_response      867.51 K        1.15 μs  ±2327.39%        1.00 μs        2.14 μs
gregvaughn_response         766.18 K        1.31 μs  ±2168.73%        1.15 μs        2.31 μs
ericgray_response           730.43 K        1.37 μs  ±1314.86%        1.26 μs        2.48 μs
chungwong_response          655.58 K        1.53 μs  ±1555.15%        1.36 μs        2.78 μs
parse_map_int_fields        567.61 K        1.76 μs  ±1491.38%        1.57 μs        3.18 μs

Comparison:
mnussbaumer_response        927.74 K
kokolegorille_response      867.51 K - 1.07x slower +0.0748 μs
gregvaughn_response         766.18 K - 1.21x slower +0.23 μs
ericgray_response           730.43 K - 1.27x slower +0.29 μs
chungwong_response          655.58 K - 1.42x slower +0.45 μs
parse_map_int_fields        567.61 K - 1.63x slower +0.68 μs

And the code used for the benchmark:

defmodule Sanitizer do
  def parse_map_int_fields(map) do
    ints_map =
      map
      |> Enum.filter(fn {k, _v} -> Atom.to_string(k) |> String.starts_with?("int_") end)
      |> Enum.filter(fn {_k, v} -> !is_integer(v) and !is_nil(v) end)
      |> Enum.map(fn {key, val} ->
        case Integer.parse(val) do
          :error ->
            {key, nil}

          {parsed, _} ->
            {key, parsed}
        end
      end)
      |> Map.new()

    Map.merge(map, ints_map)
  end

  def kokolegorille_response(map) do
    Map.new(map, fn {_k, _v} = el ->
      if has_to_be_sanitized?(el), do: sanitize(el), else: el
    end)
  end

  defp has_to_be_sanitized?({k, v}) do
    Atom.to_string(k) |> String.starts_with?("int_") and !is_integer(v) and !is_nil(v)
  end

  defp sanitize({k, v}) do
    case Integer.parse(v) do
      :error ->
        {k, nil}

      {parsed, _} ->
        {k, parsed}
    end
  end

  def chungwong_response(map) do
    parse(map)
  end

  def parse(m) when is_map(m), do: parse(Map.to_list(m), [])
  def parse([], acc), do: Enum.into(acc, %{})
  def parse([{k, v} | t], acc), do: parse(t, prepend({to_string(k), v}, acc))

  def prepend({"int_" <> _ = k, v}, acc) when is_binary(v),
    do: [{String.to_atom(k), to_integer(Integer.parse(v))} | acc]

  def prepend({k, v}, acc), do: [{String.to_atom(k), v} | acc]

  def to_integer(:error), do: nil
  def to_integer({parsed, _}), do: parsed

  def ericgray_response(map) do
    Enum.reduce(map, %{}, &do_parse_map_int_fields/2)
  end

  # Reconstruct or accumulate key value pairs into a new map
  def do_parse_map_int_fields({key, value}, map) do
    case starts_with_int?(key) do
      true ->
        Map.put(map, key, parse_value(value))

      false ->
        Map.put(map, key, value)
    end
  end

  # Boolean function returns true if field is int
  def starts_with_int?(key) do
    key
    |> Atom.to_string()
    |> String.starts_with?("int_")
  end

  # Parses the value of int field, if not an integer return nil
  def parse_value(value) do
    case Integer.parse(value) do
      {value, _} -> value
      :error -> nil
    end
  end

  def mnussbaumer_response(map) do
    reshape_map_int_fields(map)
  end

  @spec reshape_map_int_fields(map) :: map
  def reshape_map_int_fields(map) when is_map(map),
    do:
      Enum.reduce(map, %{}, fn {k, v}, acc ->
        Map.put(acc, k, maybe_reshape(k, v))
      end)

  @spec maybe_reshape(atom, term) :: integer | term
  def maybe_reshape(k, v) when is_atom(k) and not is_integer(v) and not is_nil(v) do
    case Atom.to_string(k) do
      <<"int_", _>> ->
        case Integer.parse(v) do
          {new_val, _} -> new_val
          _ -> nil
        end

      _ ->
        v
    end
  end

  def maybe_reshape(_, v), do: v

  def gregvaughn_response(map) do
    for {k, v} <- map, match?("int" <> _, Atom.to_string(k)), into: map do
      case Integer.parse(v) do
        {int, _} -> {k, int}
        _other -> {k, nil}
      end
    end
  end
end

test_map = %{
  int_a: "1",
  int_b: "not_a_int!",
  str_a: "This is a string field",
  str_b: "Another string field",
  fl_a: 0.0
}

Benchee.run(%{
  "parse_map_int_fields" => fn -> Sanitizer.parse_map_int_fields(test_map) end,
  "kokolegorille_response" => fn -> Sanitizer.kokolegorille_response(test_map) end,
  "chungwong_response" => fn -> Sanitizer.chungwong_response(test_map) end,
  "ericgray_response" => fn -> Sanitizer.ericgray_response(test_map) end,
  "mnussbaumer_response" => fn -> Sanitizer.mnussbaumer_response(test_map) end,
  "gregvaughn_response" => fn -> Sanitizer.gregvaughn_response(test_map) end
})

you can optimize it even more:
def maybe_reshape(k, v) when is_atom(k) and is_binary(v) do