Yes, but only to a limited extent
The first major constraint is forces on your vehicle. Looking at delta V maps it appears simple transfer orbits to asteroids are arriving with more than one kilometer per second of relative velocity*. To keep things simple we assume 1000 meters per second and want to limit acceleration to 2G. This means you are taking around 50 seconds and 25 kilometers to stop. 25 km of 1600 MPa Carbon fiber capable of holding 10 tonnes of vehicle at 2G works out at 11 mm across with no safety factor and masses somewhere around 5 tonnes. So suddenly we need to make that cable thicker and heavier to cope with weight of the cable...
If we make the cables shorter the force is higher so total cable mass remains the same.
This also assumes you have some sort of magic winch that maintains a steady tension running the cable out. Elastic material is not really a good idea, both because of the lower strength but because once it slows you down by stretching you need to avoid being fired back into the asteroid at close to starting velocity.
For comparison if we use a UDMH rocket with exhaust velocity of 3000 ms you'd need about 4 tonnes of propellant and have some sort of safety factor.
The second wrinkle here is material physics. If you are approaching the asteroid at 1000 ms the initial cable connection will be faster than a speeding bullet and with rather more energy involved tending to destroy fixtures and hardware. The speed of sound in the materials will also start being a factor as forces do not propagate instantly.
The third wrinkle is energy dissipation, 10 tonnes slowing from 1000 ms has 5 GJ of energy, dissipating that at 2G over 50 seconds is 100 mega watts enough to heat our 10 tonne craft from 0 to well past melting point for many materials, which is not surprising since we are doing similar 'work' to a decent sized rocket rocket engine.
Where this approach does start to make sense is where the velocities are lower and there is a chance to re-use the hardware (rather than using a rocket where each use is 'waste'), say a catapult and net to move processed metals around the asteroid.
*There are clever orbits and careful target picks that drop the arrival velocity right down, but this helps rockets and giant cables much the same.