How do I "read out" the vertex names on this graph?

You could use DepthFirstScan:

Choose the starting vertex:

svertex = Cases[VertexList[g], x_ /; VertexDegree[g, x] == 1]

{9, 33}

Extract the order of vertices:

Reap[DepthFirstScan[g, 
    svertex[[1]], {"PrevisitVertex" -> Sow}];][[2, 1]]

{9, 10, 1, 2, 4, 3, 8, 7, 6, 5, 24, 23, 25, 26, 29, 30, 37, 38, 39, 40, 43, 44, 35, 36, 31, 32, 27, 28, 21, 22, 17, 18, 14, 13, 15, 16, 12, 11, 20, 19, 42, 41, 34, 33}

We can use GraphPeriphery to get the start and endpoints, and then use these in FindPath:

periphery = GraphPeriphery[g]

FindPath[g, ##]& @@ periphery

{9,33}

{{9,10,1,2,4,3,8,7,6,5,24,23,25,26,29,30,37,38,39,40,43,44,35,36,31,32,27,28,21,22,17,18,14,13,15,16,12,11,20,19,42,41,34,33}}

If g is the graph from the OP, then you can do this in versions 9 through 11

combGrData = {EdgeRules@g, 
    GraphEmbedding@g} /. {{a_Real, b_Real} :> {{a, b}}, 
    HoldPattern@Rule[a__] :> {{a}}};
<< Combinatorica`
Graph @@ combGrData // HamiltonianPath

(* {9, 10, 1, 2, 4, 3, 8, 7, 6, 5, 24, 23, 25, 26, 29, 30, 37, 38, 39, 
    40, 43, 44, 35, 36, 31, 32, 27, 28, 21, 22, 17, 18, 14, 13, 15, 16, 
    12, 11, 20, 19, 42, 41, 34, 33}

If someone can explain the SetDelayed error I get in version 11 I'd be much abliged.