First figure out if it's a real requirement. Over what distance must this be kept to? If it's a seriously high speed signal (look at the edge-rate compared to the length of the trace) you may need to perform some simulation. The Howard and Johnson reference that is in the answer to your linked question is a great resource on this sort of thing.
If the requirement is real, then figure out much tolerance there is (your board fab can probably only get to +/- 10% of what you ask for, so take that into account).
EDIT: Looking at your part you've now posted, you are in "real requirement" territory.
80ps edges are pretty quick! The "knee frequency" at which the harmonic start to drop off rapidly is upwards of 6GHz. Assuming propagation delay is about 66% of speed-of-light, 80ps is 16mm. The rule-of-thumb is that anything longer than 1/4-1/6 of the transition time is going to need to be treated like a transmission line, which means any trace longer than a few mm!
I'd hesitate to attempt this on a 2-layer board over any difference without doing some simulation.
You'll likely have to go multilayer to get the reference plane closer to the trace which allows thinner traces to meet the impedance specification. (EDIT: As pointed out in the comments, you could do it in 2 layers, but you'll have a really thin board then!)
Alternatively, you might be able to build a coplanar waveguide structure on 2 layers which can provide the impedance you are looking for. Or maybe increase the termination resistance, which means changing the trace impedance to match, which means a thinner trace. AppCAD can help you play with parameters for these options.
Sounds like fun :)