As others have noted, "downtown" means the central business district of a city. The definition you cite saying it can mean "the lower part of a city" ... I've never heard the word used that way. Perhaps that is an obsolete or specialized meaning. But in normal conversation, "downdown" means "where the stores are".
I read once many years ago that this definition came from New York City, where the main streets are all numbered -- First Avenue, Second Avenue, etc -- and at one time the business district was the lower numbers. I don't know if that's true and I don't think it's true today, so maybe useless tidbit.
"Up" CAN mean to a more northerly place to a place of higher elevation. But that's unlikely to be the meaning here. Usually in this context one would say that the report "came in". Like, "The lab report just came in from the downtown office". Perhaps they are using "up" here as the opposite of the "down" in "downtown": it "came up from downtown". This isn't a common usage anywhere I've lived, but it might be in this particular city. Any place I've lived, we call the business district "downtown", but no one talks about going "down" there or going "up" from there. Anyone else on here heard this usage?