Are expert systems outdated, what are better alternatives to them, if any? I need to link facts to actions through rules. If a person bought soup 10 times and he is coming at midday every day, then the system should link the fact that the person bought soup so many times, using the rule "person, consecutively buying soup 5 or more times within a month is going to buy soup with 90% confidence", to the conclusion that with 90% confidence today at midday, the same person will show up, ordering soup.

Above is an example of an expert system, however based on the research I've just done, it seems like expert systems have lost their popularity, which they had in 90s-00s. The only "popular" applications are business decision making and some medical systems. I am wondering if I should use expert systems or should I look into more "advanced" alternatives? What are the obvious disadvantages to using an expert system, besides the setup cost of conducting the research?
Expert systems went through a phase of increasing acceptance and then widespread recognition of their limitations (see Wikipedia). There are many references that consider cons e.g. Problems in expert system development (Kemp, 1993).
In more recent years recommendation systems have taken over some of the function you describe in recommending products to customers. A notable development or historical event was the Netflix contest for movie recommendations which led to a burst of innovation and interest in the area. Another aspect of your question involves learning patterns of human behavior and this can be done with ANNs and similar technology.