There are different degrees of twilight that have been officially defined.
Civil twilight: When the Sun is within 6 degrees of the horizon.
Nautical twilight: When the Sun is within 12 degrees of the horizon.
Astronomical twilight: When the Sun is within 18 degrees of the horizon.
Now, it’s very hard to say exactly at what stage of twilight a particular would be visible. It will depend on atmospheric conditions, how high the star is above the horizon, and who’s lookin’.
However, I would think that Aldebaran (a 1st magnitude star) would be visible during civil twilight. On the summer solstice, the Sun is at a declination of +23.5 degrees. During the darkest part of civil twilight, the Sun six degrees below the horizon. This will occur at midnight.
[Various arcane scribblings omitted due to lack of graphical editor.]
So whatcha want is to be at around 60.5 degrees latitude (north or south, doesn’t matter.) But this is only a rough estimates. At a higher latitude, it might still get dark enough to see Aldebaran. However, the higher your latitude, the lower Aldebaran will be in the sky, the closer it will be to the horizon, where the sky is brighter.
At an RA of 4 hours, Aldebaran will be on the opposide side of the sky from the Sun at 18 hours (for southern hemisphere summer solstice) so you have that working for you.