This is a broad question and too broad for me to answer comprehensively. It should be broken down into doppler methods, transits and direct imaging; and that's before we get to questions of detecting Kuiper belts, radio emission etc.
I'll stick for the moment with what I know about detection of planets using the doppler wobble technique.
Doppler Technique
The reflex radial velocity semi-amplitude of a star for the case of a planet of mass $m_2$ orbiting a star of mass $m_1$, in an elliptical orbit with eccentricity $e$, and orbital period $P$ and with an orbital axis inclined at $i$ to the line of sight from Earth is:
$$ left( frac{2pi G}{P}right)^{1/3}frac{m_2 sin i}{m_1^{2/3}} (1-e^2)^{-1/2}. $$
A (very) detailed derivation is given by Clubb (2008).
So I built myself a little spreadsheet and assumed that all the planets were seen optimally at $i=90^{circ}$ (they could not all be seen optimally, but the smallest inclination would be about $i=83^{circ}$ for Mercury, so it doesn't make too much difference) I'll also assume the mass of Alpha Cen A is about $M simeq 1.1M_{odot}$.
The results are
Planet | RV semi-amplitude (m/s)
Mercury | $8.3times 10^{-3}$
Venus | $8.1times 10^{-2}$
Earth | $8.4times 10^{-2}$
Mars | $7.5times 10^{-3}$
Jupiter | $11.7$
Saturn | $2.6$
Uranus | $0.28$
Neptune | $0.26$
The limits of what are possible are well illustrated by a planet around Alpha Cen B, claimed to be in a 3 day orbit and with a mass similar to the Earth (Dumusque et al. 2012, and see exoplanets.org). The radial velocity semi-amplitude detected here was $0.51pm 0.04$ m/s, and some spectrographs, notably the HARPS instruments, are routinely delivering sub 1 m/s precision. Thus Jupiter and Saturn would be detectable, Uranus and Neptune are right on the edge of detectability (remember you can average over many RV observations), but the terrestrial planets would not be found (Earth detections would require precisions below 10 cm/s. Remember also that the weaker signals would have to be dug out from the larger signals due to the Jupiter- and Saturn-like planets.
However, there is a second limitation: to find a planet using the doppler method you need to observe for at least a significant fraction of the orbital period. Given that current m/s precisions have been available for only $sim 5$ years, it is unlikely that Saturn would yet have been detected.
A picture that illustrates the situation can be obtained from the exoplanets.org website, to which I have added lines that approximate where RV semi-amplitudes would be for 10 m/s and 1 m/s precision (assuming the Alpha Cen A mass and circular orbits). I've marked on the Earth, Jupiter and Saturn. Note that few objects have been discovered below the 1 m/s line. Also note the lack of planets between the 1 and 10m/s lines with periods longer than a couple of years - the recent increase in sensitivity has yet to feed through to lower mass, longer period exoplanet discoveries.
In conclusion: only Jupiter would have been so far found by the doppler technique.
Transit techniques
I'll also add a few comments about the transit technique. Transit detection will only work if the exoplanets orbit such that they cross in front of the star. So high inclinations are mandatory. Someone who is better at spherical trigonometry should use the published data for the solar system to work out how many (and which) planets transit in some highly optimal orientation. Given that the planets have orbital inclinations with a scatter of a few degrees, then some straightforward trigonometry and a comparison with the solar radius, tells you that these orbits will generally not all transit for any particular viewing angle. Indeed a number of the Kepler-discovered multiple transit systems are much "flatter" than the solar system.
The Kepler satellite is/was capable of detecting very small transiting planets thanks to its very high photometric precision (the dip in flux is proportional to the square root of the exoplanet radius). The picture below, presented by the NASA Kepler team (slightly out of date now), shows that planetary candidates have been discovered that are down to the size of Mars. However these tend to be in short period orbits because a transit signal needs to be seen a number of times, and Kepler studies this patch of sky for about 2.5 years (when this plot was produced).
So from this point of view, possibly Venus would have been seen, but none of the other planets could be confirmed.
However, there is a wrinkle. Alpha Cen A is way too bright for these kinds of studies and way brighter than the Kepler stars. You would have to build a special instrument or telescope to look for transits around very bright stars. Some of this work has been done by ground-based surveys (mainly finding hot Jupiters), but a new satellite called TESS (Transiting Exoplanet Survey Satellite, launch perhaps in 2017) will do a more comprehensive job. This is a two year mission, so might be capable of detecting the Earth and Venus (and possibly Mercury), but Mars would not produce multiple transits on this timescale.
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