JDługosz answered this great, with numbers and everything. And the question has been considered by at least two scientists, namely Crane & Westmooreland - their paper can be found in arxiv here.
Containing the black hole:
When it comes to containing it safely, well, I the authors of the aforementioned paper concluded that a BH confine itself. All that has to be done is to avoid colliding with it. I would say it is a pretty disengaged way of addressing the issue, but then again, maybe it was a no-brainer and I am the only reader left wishing for a more in-depth explanation of this facet of the problem.
Feeding the black hole:
They need not be fed at all to be efficient. If that was the case, they would likely be unfeasible as propulsion methods (it is quite likely that they are unfeasible anyway, but for other reasons than the feeding issue). Feeding a black hole with a radius of 0.9 attometers (10 to the power of minus 18) with iron, would be hard for the reason that most likely the BH would zip right through the iron bar. The average distance between bonded iron atoms is 0.7 picometer (10 to the power of minus 12), meaning we could fit a million BHs in the distance between two iron atoms in a piece of solid iron. The feeding procedure would take the term micro managing to a whole new level.
Size(s) of the black hole(s):
Which size(s) of the black hole(s) would be possible? According to Crane & Westmoreland, there is a sweet spot - and that is a BH large enough to exist and be able to accelerate/decelerate the ship during the entire voyage. Yet not larger than necessary, since BH give off Hawking radiation in an inverse proportion to their size, as mentioned by JDługosz. A bigger hole will give you less energy output. The radius of 0.9 attometers I used as an example was not a random number, but a radius that would mean that the BH would exist during 3.5 years. That is the relativistic time that a one-way trip from Earth to Alpha Centauri (~4 ly away) would take, i.e. that would be the length of the journey as measured by the clocks on board the ship, if said ship were to accelerate at 1 g for half the trip, and then decelerate at 1 g during the other half (I must presume they have assumed some value for the mass of the ship to be able to calculate this, but I couldn't find it anywhere in the text).
Smaller BHs would be useful for fast accelerations of probes or missiles. Larger BHs would be useful for longer trips. So yes, there are "sweet spots" when it comes to the size of a BH when used for propulsion. These sizes depend upon the length of the journey, the mass of the ship, etc.
But too large BHs, and the time needed to generate enough energy to accelerate the mass of the BH itself would be very long. Please see the table and the examples explained in the article for more information.
Given the size of the proposed BHs, we could most likely not ignore quantum effects. And given their density, we could not ignore the gravity effects either. Since we currently lack understanding of gravity at the quantum level, and since our understanding of black holes from both a theoretical and experimental standpoint is extremely poor, I would say that it is not possible to determine the feasibility of this propulsion method at this stage of our technological development. The question would best be readdressed when we have a workable quantum theory of gravity, if it is to remain within the realm of hard science, rather than a the field of science fiction.
EDIT: Just found a newer paper by Crane on the subject here a minute ago. Haven't read it though, but it might be relevant.