Telescopes have angular diffraction limit depending on the observed wavelength and aperture diameter.
I've read that it's possible to go beyond the limit for microscopes. But is it possible to do the same for telescopes too?
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Short answer - no, it wouldn't be possible to beat the diffraction limit with a telescope.
Longer answer - The way that microscopes get around the diffraction limit is by getting really close to what they're looking at, or into the near-field of the object. For visible light, this only exists within a couple of nanometres of the surface. The near-field contains information at all spatial frequencies (i.e. arbitrarily high resolution) but it decays away exponentially (like light in the cladding of an optical fibre or the wavefunction in the wall of a quantum well). Near-field microscopes beat the diffraction limit by converting the near-field into propagating light - into the far-field - where it can be detected and measured. The various forms of SNOM are probably the most obvious examples of this.
As telescopes can never get close to what they're looking at, the near-field will always be unavailable to them - in other words, they will only ever be using the far-field. This means that they can never gain information beyond the diffraction limit - they are fundamentally incapable of retrieving the lost information. This is also the reason that superlenses built with metamaterials won't work with telescopes - they work by amplifying the near-field.
The only ways I'm aware of for getting around the diffraction limit in the far-field is to pre-arm yourself with more information - either by cleverly structuring the light you use to take the picture, or to modify surface you're looking at with fluorescing dyes, neither of which are possible with a telescope. Maybe someone knows more though?
Fourier optics is the bit of physics that covers all the near-field/far-field information loss if you want to look in more detail.
Not in the same way as a near field microscope.
However the "Airy limit" is a totally arbitrary limit, based on the maximum of the diffraction pattern from one point being over the minima of another.
If you had a diffraction pattern from a single point and a pair of points closer than this limit - you would have a different diffraction pattern and (given enough signal to noise) you could recover the original image
The Airy disc of a point would be a set of perfect circles. A non-point, or an object with detail, would have a non-perfect circle. The irregularities in the disc can infer the shape of the object. Some company in California a decade ago got this working for dark-field imaging in telescopes and microscopes.
In microscopy you can get super-resolved images, either as mentioned before going in the near field (atomic force microscope, NSOM..), or all-optically in the far field. All the super resolution techniques in the far field involve fluorescence. In PALM-STORM you tag your sample with fluorophores which are photo-switchable and you let them blink on-off so to have their point spread functions (PSF) not overlapping at the same time, you find the center of each PSF with nm accuracy and you reconstruct the entire sample image by many different images where only few separated PSFs are present. In STED, you engineer the shape of the PSF of the excitation laser source, using stimulated emission to reduce the excitation size down to few 10s of nm. In SIM, you also use a smart illumination scheme.
So in telescopes, this kind of super resolution seems impossible without the ability to go to the near field or to engineer the emission or excitation.