There are several options for accessing the Art Newspaper:
Art & Architecture Source December 2006 onwards. A few articles, especially short pieces, may be missing. Available to SIA staff and students. Includes details, though not text, of many articles from previous years, though coverage is inconsistent. If you discover an article you are interested in this way, cross-check with SIA's collection on paper.
Video: searching within the Art Newspaper in Art & Architecture Source (2 mins)
On paper in the SIA library Issue 1 (1990) onwards. A few issues are missing, so please check the link.
Email newsletters several free newsletters, supplementary to the Newspaper itself
A personal subscription may be a worthwhile investment for you, and gives you both a paper copy and a PDF.