How important was Upton Sinclair's The Jungle to passage of the Meat Inspection Act?
In a word - Vital.
Upton Sinclair's The Jungle is one of the most noted misses in terms of mission but vital in terms of cleaning up the meat processing industry.
Sinclair wrote The Jungle in 1906 and he intended it to be a call to socialism, for the workers of America to rise up and throw off the shackles of industry leaders (as he gets to the last pages of the book, that call becomes ever more obvious and, in my opinion, clumsy).
It was a timely message (even if it didn't result in what Sinclair was looking for) - Karl Marx had written extensively about the inequalities of economic class strata in the mid 1800's and so ideas about Marxism were flowing freely, robber barons had seemingly robbed workers of the promise of prosperity for anyone who worked hard, and so Sinclair saw an opportunity to highlight the struggle of American workers by talking about the working conditions of the Chicago meat processing industry.
What happened instead of workers rising up was that the American populace saw what disgusting conditions their meat was being processed in and cried for action. As Sinclair said himself on the public's reaction "I aimed for the public's heart and by accident hit it in the stomach." That call to action led directly to the Meat Inspection Act (passed quite quickly - in 1906).