The solution to such problems is a very simple application of conservation laws. 

The first relevant law is conservation of angular momentum, which says that for a particular orbit
$$ L = mrv_{\rm tan} = {\rm constant}\ ,$$
where $m$ is the mass of the orbiting body (assumed $\ll$ the mass $M$ of the body it orbits) and $v_{\rm tan}$ is the tangential velocity component (i.e. that component perpendicular to a line between the central mass and the orbiting object) and $r$ is the distance between the orbiting object and central mass.

The second is conservation of energy which says
$$ \frac{1}{2}mv^2 - \frac{GMm}{r} = {\rm constant}\ ,$$
where $v$ is the *speed* in the orbit, which is formed from a tangential component (discussed above) and a radial component directed along a line towards or way from the central object.

In an elliptical orbit, an object in general has *both* a tangential and radial velocity component, *except* at aphelion and perihelion, where the radial component is zero and $v = v_{\rm tan}$.

Thus if we know the speed at perihelion, we can use conservation of angular momentum to work out $v_{\rm tan}$ at any other position in the orbit and we can use conservation of energy to calculate $v$ at any other position in the orbit.

If we are in possession of the speed and one of its components, then we know which way the velocity is directed.