The first SFnal solution was suggested by Robert L Forward in his book "Future Magic", where an asteroid of that size is crushed down into a sphere @ 30cm in diameter. At that point, the density is close to that of a neutron star and the surface gravity is @ 1 "g". You might find the surface a trifle small for what you want to do.
Expanding on that idea a bit, you could simply slide your sphere of super compressed matter into the centre of another asteroid. The potential problem is that the immense gravitational pull will likely disrupt the asteroid, and there will be interesting (for certain values of interesting) tidal effects due to the steep gravitational gradient.
Going even farther down the rabbit hole, cosmic strings are flaws in space-time which would theoretically have immense gravitational fields, but are (as the name suggests) string like, and the gravitational fields would follow the string rather than be point sources like a sphere. This provides some interesting effects, such as the string would provide gravity along the outer surface of a pop can shaped construct, but not at the ends (the super dense cosmic string would be a danger to anything at the end points, however). The topology of a "knotted" cosmic string might provide a more spherical (or at least lumpy) gravitational field, as well has securing the ends.
Paul J. McAuley, in his "Quiet War" trilogy, had a fictional technology of "entangling gravitons" and depositing them in the centre of an asteroid or construct to bulk up the gravity field of the object. Presumably, one could make a hoop of gravitons and create a local null gravity zone in the centre as well.
As the author, you are free to select whatever means you want (or simply handwave and don't bother to explain how this is done, much like in most SF TV shows and movies). If it isn't important to the plot, then your best bet is to ignore it.