What can the Republican Party do to increase their support with urban voters?

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Republicans here are in a similar bind to the "Democrats for small towns" strategies: they are outsiders looking in, and their motives are seen as underhand and definitely contemptuous, reliant on local proxies with a very distorted view of their place in the local fabric. You can really see this in the autumn of the infamous year of Defund the Police: a lot of people were angry at their municipal leaders for many, many reasons, but as this was noticed by the GOP and conservative leadership who tried to stir the pot, people started to go back to their leaders as they became inundated with a national message that seemed contemptuous of their way of life. Much like your Obama-Trump voter resented being told he was some kind of deranged gap-toothed bumpkin, the residents of Portland or Minneapolis or Chicago also resent the reductive stereotypes that are thrown their way.

There are lots of patterns to successful urban conservativism. My personal hobby horse has been that conservatives take a leaf from Rob Ford (a conservative with a secret power: he didn't need to lie when he said he had black friends). You could try the ethnic sectarian route (which would be repulsive, but potentially lucrative way to uproot the carefully nested multicultural nature of most Dem machines). You could try and coopt a machine in cities where anti machine reformist are threatening to win, or co-opt the reformists.  You can be positively LibDemish in how you can opportunisticly represent just about any cause. Point is you do need local people on the ground and you need to ignore the drive to nationalise every single issue.