I've recently finishing listening to Ayn Rand's "Fountainhead" on audio-book. I've read the book in paperback in the past as well as her book "Atlas Shrugged". I won't pretend to have a completely thought out approach to thinking about this subject of Rand and Christianity, but here's some thoughts.
Rand's economic theory is Laissez-faire capitalism which means allowing industry to be free from state intervention, especially restrictions in the form of tariffs and government monopolies. Rand's writings strike a particular resonance today with rampant government "stimulus plans" and unprecedented levels of government debt.
There is no singular accepted view of economics with Christians ranging from socialist to free-market capitalists. Thus, Rand's view of economics is not inherently incompatible with Christianity. If anything, the Old Testament model of economics as exemplified by ancient Israel would appear to be very much a free market system with no centralized government.
Rand rejects Statism, at least in economics. She does grant there is a role for government in keeping order, ie, the Constitutional limits provided by the original Constitution (national defense).
Few Christians would explicitly admit to be Statists. The history of early Christianity often pitted Christians against the state resulting in the deaths of untold Christians at the hand of the state. Many view the book of Revelation as a discourse against the coercive power of the state. However, many Christians also look to the state for their economic security (social security and unemployment come to mind particularly). The state has created a dependency class which, once started, is notoriously hard to dismantle. In the US, we need to look at how other states, such as Russia, have handled moving away from Statism.
Unlike the caricatures of her view, Rand doesn't oppose the idea of individual charity. Quite the contrary. Her opposition is more to the implementation of charity than the notion. Charity can't be forced or done to lord one's superiority over the person who is the object of charity. The person doing charitable work should do their work because they get something out of it, rather than the persons who receive the charity.
In Fountainhead, Howard Roark builds a government project, not to help the people in need, but because it's an architectural challenge which he believes he can meet. It's fine that people are helped (although are they really helped the book asks), but should not be the motive of the one doing the giving.
Similarly, Christians would oppose charity which is done for the sake of lording superiority over the one that is served. Overlapping the issue of economics, there are Christians who would see the idea of taxes funding charity as usurping individual charity, ie, not the proper function of government. In a society where people are taxed at about 1/2 of what they make, there's not much left to give in a charitable manner.
Rand was an atheist/agnostic. She was an anti-supernaturalist. Rand enshrined reason as the highest value, functioning effectively as a god in her view.
Christians are supernaturalists in principle, but rarely in practice. Although Christians pray for help from God (particularly in times of need), they don't have much expectation of day-to-day intervention in their lives. In the realm of reason, Christians have a belief that reason has an origin with God, in His nature and in the Imago Dei which is given by God to man as part of His creation.
Who is Jesus?
The key notion of Christianity is that God became man in the person of Jesus of Nazareth. Rand's main characters (Roark and Galt) are messianic in the sense that they are very strong individuals who are set out as prototypes of authentic human behavior against a backdrop of a mankind which is itself inauthentic. They don't seek to gather followers, but because of their authentic testimony of their humanity, they can't help but gather disciples.