Clipping image with Basic (ArcView) level license of ArcGIS Desktop?

You can clip a raster to an irregularly shaped polygon (following the polygons geometry) using the tool you tried before, Clip.

Scrolling down a little further in your screenshot, click the checkmark to enable Clipping Geometry (see image). You should have this available at v9.3; This screen is from 9.3.1

Could you create a large buffer outside the object, and then fill this buffer with a white color?

It sounds like you just need a graphical way of hiding the image, rather than something that allows you to perform a calculation.