The Britain Express website tells us:
A Victorian Gothic (Catholic) cathedral in French Gothic style, immediately beside Arundel Castle. The Cathedral was begun by Henry, 15th Duke of Norfolk, in 1868, to serve as a Catholic parish church for Arundel. It was completed in 1873.
The Duke employed as his architect Joseph Hansom, inventor of the Hansom cab. While this might seem an odd choice, Hansom carried out his brief brilliantly, creating a soaring church faced with Bath stone.
The Duke wanted a church to rival his own magnificent castle, and Hansom obliged, creating one of the finest examples of Gothic Revival style in England, with slender clusters of pillars rising to beautifully vaulted ceilings soaring high overhead.
The original design called for a spire over the north porch, but this idea was never carried out, leaving the church with a rather peculiar outline. The most striking feature of the cathedral is the west front, with a huge rose window filled with colourful stained glass.
St Philip Howard
The cathedral was originally dedicated to St Philip Neri, then in 1965 this was extended to Our Lady and St Philip when the church was raised to cathedral status, and finally in 1973 the recently canonised St Philip Howard was added to the dedication. Howard was the 20th Earl of Arundel, a Catholic recusant who was imprisoned in the Tower of London by Elizabeth I and died of dysentery there in 1595.
He was buried first at the Tower, then reburied in the Fitzalan Chapel in the castle grounds. Finally in 1971 he was reburied again, inside the cathedral, where you can see his memorial, with a sculpture of his faithful dog, who kept him company throughout his decade-long stay in the Tower.
The cathedral's website tells us:
The Cathedral was founded by Henry 15th Duke of Norfolk, whose old established family own extensive estates around Arundel, and the building was completed on 1st July 1873, to designs of Joseph Hansom, inventor of the Hansom cab.
The original dedication was to St Philip Neri, founder of the Oratorian religious order. When the Diocese of Arundel and Brighton was created in 1965, the church of St Philip at Arundel was made its cathedral and rededicated to Our Lady and St Philip.In 1973 there was a further change in dedication to the recently canonised martyr St Philip Howard.
The cathedral is Grade I listed with the entry at the Historic England website telling us:
Designed 1868-9. Built 1870-3. Architect J A Hansom. Ashlar walls and vaults. Pitched slate roof. Picturesque Gothic composition with details from later C13 sources. Stylobate surrounding cathedral on north and west sides, interspersed with ashlar piers: square bases, octagonal tops, and pyramidal caps with foliate finials. Roman Catholic Cathedral of St Philip Neri forms a group with No 2 Parsons Hill.
Wikipedia has an article about the cathedral that tells us:
The Cathedral Church of Our Lady and St Philip Howard is a Roman Catholic cathedral in Arundel, West Sussex, England. Dedicated in 1873 as the Catholic parish church of Arundel, it became a cathedral at the foundation of the Diocese of Arundel and Brighton in 1965. It now serves as the seat of the Bishop of Arundel and Brighton.
The cathedral's location, construction, design, and dedication owe much to the Howard family, who, as Dukes of Norfolk and Earls of Arundel are the most prominent English Catholic family, and rank first (below the royal family) in the Peerage of England. Since 1102 the seat of the Howards' ancestors has been Arundel Castle.
In 1664 Roman Catholic worship was suppressed in England by the Conventicle Act and all churches and cathedrals in England were transferred to the Church of England. With the Catholic Emancipation Act of 1829 the foundation of Roman Catholic parishes became again legal.
In 1868 Henry Fitzalan-Howard, 15th Duke of Norfolk commissioned architect Joseph Hansom to design a new Roman Catholic sanctuary as a suitable counterpart to Arundel Castle. The architectural style of the cathedral is French Gothic, a style that would have been popular between 1300 and 1400—the period in which the Howards and the Dukes of Norfolk rose to national prominence in England. The building is Grade I listed, and regarded as one of the finest examples of Gothic Revival architecture in the French Gothic style in the country.
The church was originally dedicated to Our Lady and St Philip Neri, but in 1971, following the canonisation of Philip Howard, 20th Earl of Arundel and the reburial of his relics in the cathedral, the dedication was changed to Our Lady and St Philip Howard.
In 2002, Elizabeth Stratford was appointed Organist and Master of the Choristers of the cathedral, becoming the first woman to hold the post of Director of Music in an English cathedral. Stratford was educated at St. Joseph's Catholic College, Bradford and then at the University of Huddersfield winning scholarships for voice, composition and organ from the RCO and other trusts. She studied at the University of Leeds with Gordon Stewart (organ), Simon Lindley (choir training) and Philip Wilby (composition). She succeeded Alistair Warwick as the Organist and Director of Music of the cathedral, and she also teaches piano at Brighton College.