How feasible would it be to get to a planet orbiting directly opposite Earth?
Venus has roughly the same gravitational potential as earth and the Venera probes successfully and softly landed there. Similarly, the Apollo space craft landed safely on earth on return. In both cases aerodynamic friction with the atmosphere can be used to get rid of excess velocity in order to enter orbit and eventually land. Thus landing will work cheaper, if there is atmosphere. But at the expense of much more fuel also without (see moon landing again... you'd need stronger descent rockets than Apollo for an exclusively rocket - driven landing) 

Getting there is not a big deal either. Look at the Stereo mission of two satellites which trail earth's orbit more and more to get a better understanding of the sun, gaining more and more separation from earth while (nearly) staying at 1 AU: You get to another position in earth's orbit by just lifting your orbit slightly and waiting for earth to go and the other to catch up (or go slightly towards the sun and be faster yourself).