What prevents a neuron from activating an incoming synapse? As is known, a neuron can have incoming and outgoing synapses. When a neuron fires, what does prevent that only outgoing synapses are activated?
Basically there are two reasons. First, (most) synapses are asymmetrically designed in such a way that synaptic transmission only works in one direction. Second, action potentials usually run in one direction only, from the axon hillock down the axon.

Saying that a neuron fires means that an action potential (AP) is effected, a series of  depolarisations along the axon which constitute a signal that moves down the axon, since APs are usually initiated at the axon hillock. Neural signal flow thus generally is directed orthodromically, i.e. along the axon and away from the soma.

A synapse is constituted partially by the presynaptic neuron, partially by the postsynaptic neuron. It is the axon terminals, located at the distal end of the presynaptic neuron's axon, which can carry a neural signal further over the synaptic cleft to effect an AP in the postsynaptic neuron. This generally works in one direction only – e.g. the presynaptic part of the synapse can release neurotransmitters, and the postsynaptic part of the synapse has receptors for the neurotransmitters, not the other way round (mind that this does not imply that the postsynaptic part of the synapse cannot have any effect on its presynaptic counterpart).

Since the propagation of the AP along the axon basically works through causing neighbouring parts of the axon to depolarize, it would be possible that at some point a second AP is generated which runs back into the direction the original AP came from. However, this is prevented by afterhyperpolarization (also called refractory period): the membrane potential falls below the usual resting potential and thus prevents this running back up of a second, antidromic AP.

The mechanism of afterhyperpolarization is largely responsible for unidirectional propagation. However, it has to be emphasized that this only applies if the AP is initiated at the axon hillock in the first place. If an action potential forms at some point down the axon, it is indeed possible that two APs are generated, one moving down the axon, another one moving upwards towards the soma. This is called neural backpropagation, and while it constitutes an actual phenomenon in the nervous systems, there is not much clarity yet about its (possible) functions. However, the above mentioned unidirectional design of chemical synapses prevents this signal to effect synaptic transmission.

Another phenomenon that can be said to reverse the »normal«, unidirectional flow in neurons is that of electrotonic potentials (but they don't effect anything like antidromic APs), and finally an autapse is a synapse from a neuron's axon onto its own dendrite, so that in some sense it is correct to say that a neuron with an autapse activates an »incoming synapse« (however not in the sense the question seems to imply). Finally, it should also be mentioned that in contrast to chemical synapses, electrical synapses work in both directions (notwithstanding the above).