You cannot rotate text inside a text frame. But...
To rotate the bounds of a text frame without also rotating the text inside,
- Select the text frame.
- Switch to the white arrow (the "Direct Selection Tool").
- Use "Select All". In the context of the white arrow, this means the points of your text frame's frame.
- Rotate any way you prefer: using the Rotate tool, entering a value into the Angle box in the Control bar, pressing the 90° Clockwise/Anticlockwise buttons, or use the Object > Transform > Rotate menu.
This means you have to adopt a 2-step approach to see just your text rotated "in place". Make sure the Reference Point proxy in your Control Bar has its center selected. Rotate the frame per above instructions. Select the object with the black arrow (toggling back to the black arrow is enough), and rotate it the same amount in the negative direction. If you are rotating by 90° steps, you can use the buttons for that in the Control Bar.
It would have been better to draw object and text as two disconnected objects, so you can change one's orientation without changing the others'. Even if you have grouped them together, you can select only the frame object to rotate just this with Object > Select > Content (there are also buttons for this in the Control Bar).
To easily change text formatting overall, you can apply a Paragraph Style to all of your text and change only this when required.