Try setting `TextPadding -> True`.  It does more or less the same as the builtin  [`ContentPadding`](http://reference.wolfram.com/language/ref/ContentPadding.html).  It is described in SciDrawGuide.pdf.

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What `ContentPadding -> True` does, in simple terms, is that it makes the enclosing box have a vertical size equal to a full line height.  Otherwise the box has the same height as the enclosed character/text.  SciDraw positions textual elements wrapped in boxes.  For good alignment it is necessary that all boxes have the same height.  This is done by enforcing a minimum line height.  [MaTeX](https://github.com/szhorvat/MaTeX) also has such an option, precisely for dealing with this kind of problem.

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Example:

    << SciDraw`

    labelFun[x_, _] := 
      Module[{frac = Rationalize[x/Pi] Pi, num, den},
        num = Numerator[frac];
        den = Denominator[frac];
        If[den == 1,
          num,
          StringForm["``/``", num, den]
        ]
      ]

    Figure[
     FigurePanel[{FigGraphics@Plot[Cos[ωt]^2, {ωt, 0, 2 π}];}, 
      XPlotRange -> {0, 2 π}, 
      XTicks -> LinTicks[-2 Pi, 2 Pi, Pi/2, 4, TickLabelFunction -> labelFun],
      TextPadding -> True,
      XExtendRange -> 0.02, YExtendRange -> 0.04
      ],
     CanvasSize -> {5, 3}]

I'm not sure what the best way is to get correct spacing around the `/` sign, but the following works:  replace the string in `StringForm` with `"\!\(TraditionalForm\```/``\)"`.  This is a string representation of boxes.  I obtained it by writing a single character into a string, selecting it, then pressing Command-Shift-T to convert it to TraditionalForm.  This created a traditional form environment within the string. Then I deleted the character and typed what I wanted instead.

[![enter image description here][1]][1]

To set things such as font size, etc. use options such as `FontSize`, `TickFontSize`, `XTickFontSize`, etc., depending on how specific you want to be.  It is not necessary to use `Style` here.  These can also be set at the level of the `Figure`, as these options are inherited.  Or you can use `DefineStyle` and re-use it across several figures.  That's what I usually do when using SciDraw.

You may also be interested in the `*TextNudge` options, e.g. try `XTickTextNudge -> -2` to position the ticks further from the frame.

  [1]: https://i.sstatic.net/FgfRH.png