Usually, when that comes up, the city is selected, and you can see what your city is producing.  Now, I say Usually, because it does mess up a lot.

In other words, short answer is no. Many players have taken to naming their cities in a way that will help remind you (for example put an "A", "G", or "T" in front of the name to stand for Arcane, Gold, or Tech).

1 can be 1.3, 2 can be 1.9.  If you look a the city screen, it gives you the actual number.

i've taken to writing down a list of my most important towns atm on some paper. I usually take the gold bonus unless i have the tech or arcane  +5 resource available. I just realized recently while looking at the individual city info cards that the bonuses all stack - i assumed that the level up bonus replaced the previous ones so this changes my strat a bit for the future - i might have cities that do +10 arcane, +20 tech and +30 gold for example.

you know, if units did get xp every turn - then the sentinal units might become worth it if they got stronger as the game progressed. I'm going to try picking them in my current game. I hit the 50 tile limit for the first time in a city that i captured from an ai - it was his starting city and it had 3 gold mines, a tech and an arcane  resource plus a farm i decided to build my palace there and the mint of revanna so it's making 80+ gold , it might hit 200 later on. bizarre luck at times. bizarre city.

All you need is gold. Check gold every time for every city, it's the best choice.

It would be nice if you could put the 'level up' thing to the side so you can look at the city first (or even just put a 'look at city' option on the pop-up)

better solution: ditch level up bonuses so people don't have to worry about these sorts of things.

if you want cities to be able to specialise as they level up, assign these specialisations to buildings that require a certain level (and that you can only choose one of from a selection). that way players can do the specialising as and when they choose, and all the info is contained in one place: the buildings list.

level up bonuses are a clunky, half hearted way to make people care about levelling up. it smacks far too much of the "+arbitrary unexplained percentage" nonsense that we had too much of in gal civ 2. we have buildings, levels, and soon we're getting specialist slots as well. why keep an underdeveloped, tacked on mechanic that can be managed much better by the other systems?

I agree. Thirded.

These days I name my cities in a manner that tells me what the bonus they need is when this screen comes up. Useful for the occasional time when the screen doesn't centre on the city.

Yep!

Don't require patience tbh. In the past I'd actually used to give my cities cool names based on some little theme that I devised for my empire, but the whole bonuses and the screen not centring on the city was a pain.

So now all cities are either: Food City, Tech City, Gold City and Magic City. End of story. Problem solved........though they are very boring city names; the price I pay for the ease of assigning the bonuses properly =P

This. Money makes the world go 'round.

But it would be nice to have an alternative notification to the pop-up system, one that does not obscure the screen, and lets us handle the alerts at our convenience. 

Yeah, and seventh'd.