I considered starting with a level dip into Cleric with Exploration subdomain, then becoming a Druid with it afterwards. Does this method work or is there any way to acquire this domain as a single-classed Urban Druid?
You can get it as a cleric, but it will be a cleric domain only
Multiclassing is normally open to all, so as long as you can justify it within the character concept (and cleric/druid is generally not hard), taking one level of cleric should be no problem.
As a cleric, you could choose the Exploration domain as normal, and get all the features that a level 1 cleric gets. However, it's a granted feature of that class, and so only scales with levels in that class.
When you subsequently gain druid levels, you get a new domain (as per Urban Druid), your Druid domain will use your druid levels for their powers only:
When determining the powers and bonus spells granted by this domain, the druid’s effective cleric level is equal to her druid level.
As domains depend on class level, not character level, the two domains will be handled seperately, so your exploration domain will only grant the level 1 powers, and you can only use the level 1 domain spell, as a level 1 cleric spell.
Unless you really want the level 1 Domain power, this is a bad solution - you'll give up a whole level of druid for what will eventually be a weak power and some low-level spells, most of which you have anyway.
As far as I've been able to find, there are no ways in Pathfinder of obtaining an additional domain in your current class (let alone one the class doesn't allow), and there is no way to "combine" domains of different classes.
Of course, there is always the option of selling the idea to the GM, and being allowed to pick the Exploration domain instead of the regular Urban Druid ones.
Unfortunately, taking a level of Cleric will give you the Domain you want, but only for the cleric class. There's a solution, though:
Use magic devices
Specifically, acquire an Icon of Aspects and either a very high Use Magic Device skill or a single level in cleric (but do not take the exploration domain if you take the cleric level or the item won't work for you regardless of UMD!). The item lets you swap out the granted powers of one domain you have for any domain you don't have, subject to potentially slightly worse level restrictions on the abilities you swap. Depending on how you read the item, you may even be able to swap out some of your abilities while retaining others, but that's definitely ambiguous. The item costs 5,500 gp, is slotless, and you only need to have it when you prep spells.
That takes care of everything but the Exploration Domain's bonus spells, since all three of the abilities for the Exploration Domain are 'granted powers'. If you want those, the best way is to be a Samsaran. Barring that, you can add them to your druid spell list via levels in the Arcane Savant prestige class (at the cost of one level of spellcasting progression and losing progression of your druid features, or two feats and progression of your druid features), or via Pages of Spell Knowledge (requiring more UMD). You can get the 1st and second level domain spells cheaper via wayfinders if you're using the second method of resonant power generation for ioun stones.