I do know I need to upgrade the box but I need to install my electric dryer immediately for use. I have the breaker ready to install but have come across the problem of I see no neutral bus bar for the neutral wire. What can I do or can I install a bar on my own?

That bar IS your neutral bar.

What is that panel to the left?
Where are those extra wires going from the right?

We also have a glass fuse panel connected. Now can I attach both the ground & neutral wires to the bar? I though it was a ground bar because of the green lug screw.

That bar is bonded to the panel box so it acts as both the ground and neutral bar.

If this panelboard is a secondary panelboard, then you must have a four wire feed to it and you must not connect Neutral and Ground to the same bar. So the secondary panelboard else needs a ground bar.