Returning a single row

The problem is the return type. The method you are in is expecting you to return a string, but reader["col_1"] is an object. I suggest returning reader["col_1"].ToString() or Convert.ToString(reader["col_1"]).

To me it seems, you don't want a single row, only a single value:

SqlConnection sqlConnection = new SqlConnection("Your Connection String");
SqlCommand cmd = new SqlCommand();
Object returnValue;

cmd.CommandText = "SELECT TOP 1 col_name FROM Customers";
cmd.CommandType = CommandType.Text;
cmd.Connection = sqlConnection1;

sqlConnection.Open();

returnValue = cmd.ExecuteScalar();

sqlConnection.Close();

return returnValue.ToString(); //Note you have to cast it to your desired data type

This is how I would style (and fix) the code:

using (var connection = new SqlConnection(ConfigurationManager.AppSettings["connection"]))
using (var command = new SqlCommand(@"select top 1 col_1, col_2 from table1", connection))
{
    connection.Open();

    using (var reader = command.ExecuteReader())
    {
        if (reader.Read()) // Don't assume we have any rows.
        {
            int ord = reader.GetOrdinal("col_1");
            return reader.GetString(ord); // Handles nulls and empty strings.
        }

        return null;
    }
}

Using the index reader[] will give you object types, these need casting. However, I hardly touch that style and always favour the slightly more verbose, but more robust use of ordinals and asking for types in a strongly-typed manner.

If you only need the value in the first column of the first row, you can use ExecuteScalar instead, again this returns an object that can be cast and doesn't need a reader:

using (var connection = new SqlConnection(ConfigurationManager.AppSettings["connection"]))
using (var command = new SqlCommand(@"select top 1 col_1, col_2 from table1", connection))
{
    connection.Open();

    var result = command.ExecuteScalar();
    return result == null ? "" : (string)result;
}

reader["col_1"] returns object.

You want something like reader.GetString(reader.GetOrdinal("col_1")).

Edit -> I just wanted to add a note here that, in addition to the concerns others have raised, a SELECT TOP without an ORDER BY can give you random results based on schema changes and/or merry-go-round scans.