I'm aware that 下町 means downtown, and 職人魂 means craftsmanship, but what does it mean when the two are brought together?
職人魂 is simply "soul/pride of a craftsman" or "craftsmanship", but I think interpreting 下町 as "downtown" is misleading. 下町 refers to the following concept described in Wikipedia:
Shitamachi is the traditional name for the area of Tokyo including today the Adachi, Arakawa, Chiyoda (in part), Chūō, Edogawa, Kōtō, Sumida, and Taitō wards, the physically low part of the city along and east of the Sumida River.
Generally speaking, the term Yamanote has a connotation of "distant and cold, if rich and trendy", whereas "Shitamachi people are deemed honest, forthright and reliable".
The inhabitants of Yamanote were thought of as espousing modernising ideals for their country, based on Western models. The people of Shitamachi, on the other hand, came to be seen as representatives of the old order and defenders of traditional cultural forms.
Shitamachi is associated with petty entrepreneurs, restaurant owners, small shop-owners and workshops, while Yamanote suggests the business executive, and the office worker.
After a long period of post-war economic decline, in the 1980s a "Shitamachi boom" emerged, with increased interest in and celebration of Shitamachi culture and history, in particular during the Edo period. Shitamachi culture is thus depicted as more authentic and traditional (while Yamanote Tokyo is the present and future), and its valorisation has been described as a refuge from the rapid modernisation of the economic boom years.
Until relatively recently, 下町 was mainly associated with the good old lifestyle of people in the past. But after the success of Shitamachi Rocket, many people started to celebrate "shitamachi-ism" even in the context of industrial manufacturing. 下町職人魂 is an opposing concept of the automated, highly-standardized, mass-production industry which is sometimes seen as inhumane. A stereotyped 下町職人 works at a smaller workshop and relies more on things like humanity, traditional apprenticeship, so-called monozukuri spirit, manual work and individual skills. It is widely believed that some 下町 workers are able to make best-quality products (even better than those made by large companies) owing to their sincerity and individual skills.
A better way to understand that phrase is 下町+職人+魂. As you said, 下町 means downtown/old town, 職人 means craftsman and 魂 means spirit/soul. So, putting them all together, you get something along the lines of the
old town craftsman spirit. I imagine this is talking about some unique characteristics of the craftsmen who work in the old town.