Try setting TextPadding -> True. It does more or less the same as the builtin ContentPadding. It is described in SciDrawGuide.pdf.
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 also has such an option, precisely for dealing with this kind of problem.
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.
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. I tend to find the SciDraw default too tight.