As a partial answer:

Note that you can simplify your Meijer-$G$ expression a fair bit to make it amenable to asymptotic analysis. Using the absorption and flip identities, as well as switching to the generalized $G$-function, we have the equivalent

$$g(t)=\frac{2}{\sqrt\pi t}-G_{3,0}^{0,3}\left(\frac2{\pi t},\frac12\middle|\frac12,1,\frac32\right)$$

or in Mathematica form:

g[t_] := 2/(Sqrt[π] t) - MeijerG[{{1/2, 1, 3/2}, {}}, {{}, {}}, 2/(π t), 1/2]

This should now be more amenable to asymptotic analysis; your limit of interest is now equivalent to

$$\frac{2}{\sqrt\pi}-\lim_{t\to\infty} t\,G_{3,0}^{0,3}\left(\frac2{\pi t},\frac12\middle|\frac12,1,\frac32\right)$$

Plotting still needs the assistance of a high WorkingPrecision setting, however:

Plot[g[t], {t, 1, 100}, WorkingPrecision -> 45, PlotRange -> All]

Unfortunately, I quickly run into trouble when I directly use Series[] or Limit[]; this seems to be an inherent problem of the underlying Meijer-$G$ implementation when the arguments are too large or too small for a given set of parameters. Altho the terms in Series[t MeijerG[{{1/2, 1, 3/2}, {}}, {{}, {}}, 2/(π t), 1/2], {t, ∞, 0}] give rise to Indeterminate terms upon direct evaluation, some experimentation with ridiculously high precision evaluations seems to indicate that the coefficients do drop to zero remarkably quickly, e.g.

Block[{$MaxExtraPrecision = 100}, 
      N[MeijerG[{{-(1/2), 0, 1}, {}}, {{}, {}}, 1*^-5, 1/2], 1*^4]] // N[#, 20] &
   3.8231551601852422131*10^-2802

(Note the exponent.)

Some more analytical work will be needed for a rigorous proof, which I might append to this answer if I find the time.

As a partial answer:

Note that you can simplify your Meijer-$G$ expression a fair bit to make it amenable to asymptotic analysis. Using the absorption and flip identities, as well as switching to the generalized $G$-function, we have the equivalent

$$g(t)=\frac{2}{\sqrt\pi t}-G_{3,0}^{0,3}\left(\frac2{\pi t},\frac12\middle|{{\frac12,1,\frac32}\atop{\text{—}}}\right)$$

or in Mathematica form:

g[t_] := 2/(Sqrt[π] t) - MeijerG[{{1/2, 1, 3/2}, {}}, {{}, {}}, 2/(π t), 1/2]

This should now be more amenable to asymptotic analysis; your limit of interest is now equivalent to

$$\frac{2}{\sqrt\pi}-\frac2{\pi}\lim_{t\to\infty} G_{3,0}^{0,3}\left(\frac2{\pi t},\frac12\middle|{{0,\frac12,1}\atop{\text{—}}}\right)$$

Plotting still needs the assistance of a high WorkingPrecision setting, however:

Plot[g[t], {t, 1, 100}, WorkingPrecision -> 45, PlotRange -> All]

Unfortunately, I quickly run into trouble when I directly use Series[] or Limit[]; this seems to be an inherent problem of the underlying Meijer-$G$ implementation when the arguments are too large or too small for a given set of parameters. Altho the terms in Series[MeijerG[{{0, 1/2, 1}, {}}, {{}, {}}, 2/(π t), 1/2], {t, ∞, 2}] give rise to Indeterminate terms upon direct evaluation, some experimentation with ridiculously high precision evaluations seems to indicate that the coefficients do drop to zero remarkably quickly, e.g.

Block[{$MaxExtraPrecision = 100}, 
      N[MeijerG[{{0, 1/2, 1}, {}}, {{}, {}}, 1*^-4, 1/2], 1*^4]] // N[#, 20] &
   1.4065564604009371892*10^-603

(Note the exponent.)

Some more analytical work will be needed for a rigorous proof, which I might append to this answer if I find the time.