This is the photo section, not a technical forum but I'll point you in the right direction...
There is a known hot start problem with 98-99.5 A4 TDIs. Pull the connection to the coolant temp sensor and start the car while it's hot. If it fires right up you have this problem (this test will probably cause a check engine light).
There are a couple of solutions - most popular is getting a good tune. The tuners change the fuel maps to start injecting fuel below 250RPMs when the car is warm (70 degrees, I think). The other, cheaper solution is the hammond mod. It's a switch or relay that opens the temp sensor connection. I wired a switch up on the dash and it makes a big difference. Do some searching on here for the directions.
Also make sure your battery is good and connections are clean. If something is preventing the engine from turning over fast enough, fuel will not be injected.