4mm Festo compatible tube may be a tight fit over the "needle" but it looks like the stuff you have is too loose for your needle.

The Rolls Royce solution is to take a piece of hex cross section brass rod and drill it out in a lathe. On one side drill it to the diameter of your football needle and on the other drill it to 4.1mm. Tap the 4.1mm side with a 5mm metric tap - Festo fittings that wil screw securely onto this are readily availabe.

The solder (preferable hard solder) the other side onto your "needle". (Even better to cut a schrader valve thread onto the outside)

Otherwise you could use 3mm tube and a 3mm-4mm Festo adapter. If the 3mm tube won't go over the needle then you could connect via a 2 way nipple and standard airline. In general standard airline (designed for nipples) doesn't work very well with Festo fittings -although Festo 3mm tube works fine on an airline nipple.