How to save datatable first column in array C#

I suggest to use LINQ to DataSet for querying DataTable:

datatable1.AsEnumerable().Select(r => r.Field<string>("Name")).ToArray();

Mode classic

List<String> stringArr = new List<String>();

// Classic version :-)
for( int a = 0 ; a < datatable1.Rows.Count ; a ++)
{
    stringArr.Add(datatable1.Rows[a]["Name"].ToString());
}