I do not believe that your second `DSolve` gives the inverse of `y`.  In any case, it is easy to plot the inverse.  With Ulrich Neumann's answer as a starting plot, use

    ParametricPlot[{{(y[x] /. sol)[[2]] /. {y1 -> 3/10, a -> 13/10, b -> 7/10}, x},
                    {(y[x] /. sol)[[1]] /. {y1 -> 3/10, a -> 13/10, b -> 7/10}, x}}, 
                    {x, 1, 10}, AspectRatio -> 2, AxesLabel -> {y, x}]

[![enter image description here][1]][1]

However, if you really do wish to plot the result of the second `DSolve`, designate it as `soli` and extract an equation for `t` as a function of `x`.

    soli[[1, 1, 2, 1]] == soli[[1, 1, 2, 0, 1]][x]
    Sqrt[10] # & /@ % /. {a -> 4, b -> 0, C[1] -> 1, C[2] -> 0}

    (* t == (Sqrt[5/2] Sqrt[1 + x^10] Hypergeometric2F1[1/2, 3/5, 8/5, 
           -x^10])/(3 Sqrt[10/x^12 + 10/x^2]) *)

(I have chosen the constants for convenience.  Other values also work.)  Then, use `ParametricPlot` as before.

    ParametricPlot[{%[[2]], x}, {x, 0, 1}, AspectRatio -> 1, AxesLabel -> {t, x}]

[![enter image description here][2]][2]

There are, of course, other ways to plot inverses of functions.  The method shown seems simplest to me

  [1]: https://i.sstatic.net/uDK8F.png
  [2]: https://i.sstatic.net/egsYK.png