I recommend not "investing" in a city-state unless you can get some influence for free to start out. Killing Barbarians or doing another quest for them will get your foot in the door, and let the $250 you have to spend periodically give you a lot more value. For instance, that $250 will be keeping you above the "Allied" line instead of just barely making you friends with the city-state.
Likewise, as mentioned above, the Patronage tree will give you significant bonuses to your influence over city-states.
It seems that the design is such that in the early game you won't be able to guarantee a city-state ally without committing a lot of cash.
To elaborate (not 100% sure on all these numbers but the overall point is still valid):
If you spend $250 to become Friends with a city-state, starting from 0 influence, you'll get 35 influence, making you barely Friends (30 influence). This means you'll be friends for 5 turns, at which point you'll need to spend another $250 to be friends again. You probably don't have another $250 that soon, though, so the influence runs all the way down close to zero before you're able to spend another $250, which again only gives you a few turns of friendship. This is obviously unsustainable.
If instead you perform a mission for the city-state and get 30 or 40 influence to start with, every time you spend $250 you'll get to use all 35 points of that influence before needing to spend another $250.
Similarly, you'd be better off saving your money to start with $500 worth of influence than buying $250 and not being able to afford more right away.
Long story short: if you're already friends, you get more value for your money. Likewise, if you're already allies, you get even MORE value for money you spend to sustain at that level.
The diplomacy summary that you get from clicking the scroll in the upper right will show you all the civs and city states at once, as well as what resources they have available, how much gold they have, and how much gold per turn they are making.
Best Answer
There is a way, but it likely involves too much work.
To answer your question directly - which assumes you settling near that area is not an option - there are two ways to do this:
1) Expand your borders:
Eventually, your borders will keep growing even past the 3 workable tiles. I believe the maximum is 5, so if it's close - you'll eventually get it.
In addition, even if it's 20 spaces away, you can still get it. To do this, you'll need a bunch of Great Generals. These guys can simulate what was known in the past as a culture bomb. Place them on the border of where you want to expand, and consume them to create a citadel; this will expand your border 1 tile in all directions (it will even replace other civilizations borders with yours!). With enough Great Generals, you can get anywhere you want (and your borders can look quite odd - too!)
2) Gifting 'glitch'
I'm not sure if this is a glitch or not, but this also involves great generals. To do this, annex a city with tiles you want within its borders. Then, consume a great general over the tiles you want to keep. When your city stops revolting, trade it to a civilization - they'll get the city, and all the unhapiness - while you keep the citadel tile and its surroundings.
Often this results in you having small, uncontected patches of land all over the place.