The Alex, as it's known locally, has always been a bit of a mystery to me. It is a very large building purported to have been built c1830 but I can only find records of a landlord from 1871. It may have been named after Queen Alexandra (1844-1925), wife of Edward VII.
It was built as a hotel to accommodate the growing influx of tourists from London wanting to spend time at the seaside with the advent of train travel in the mid 1840's to Worthing.
Despite its location some distance from the high street in a then underdeveloped area, we can clearly see in the old pictures a bus timetable on the wall. It is thought to have been remodelled in 1898.
Notice in the later picture below the corner door has gone.