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James,
Maybe my earlier response was a bit terse and therefore limited.

The general answer to your question "Is this probe a good match for my scope?" should be "no". Chuck's answer and mine show you why:
You will not in practice achieve the "promised" 400 MHz bandwidth with that combination. But, neither will you with the Tek-specified P6137. That does not mean there's no observable difference between the two under any circumstances. Under specific test circumstances the difference is clear.

If you purchase a P6139A with the intention to observe and measure over the full bandwidth (bw) of the 2465B 'scope, you'll be disappointed. In fact, *no* high-impedance passive probe will do, as Chuck explained. One might add that the source impedance used for specifying scope bw is 25 (!) Ohm. Not many real-life signal sources fit that requirement!

At frequencies below 100 MHz or edge speeds slower than ca. 15ns the P6139A will provide acceptable fidelity. Most people use their 'scopes in that range and it makes no real difference if you use a P6139A, P6139B or a P6137.
Remember that for looking at edges, it's not the frequency but the edge speed that determines the speed requirements. For faster speeds either a fast FET probe or a Lo-Z probe with 50, 500 or 500 Ohm is a better choice. Get a copy of Tek's "ABC of Probes" or Doug Ford's "The Secret World of Oscilloscope Probes" and read all about it.

As said, for things like audio, Arduino projects, shortwave Ham and the like, a P6139A or almost any other "HF" quality passive probe will make you happy. That's where Chuck's P6105, 6, 7 choice comes in. They can often be had at low prices. If you get one of those, try and get a low-inductance ground clip/wire, unless you only do audio. The standard ground wire (4 - 6 inches) produces huge ringing, even at frequencies below 100 MHz. And choose an attenuation of 1:10, *not* 1:1. If you want to know why, read the publications mentioned above.

For anything faster, go Lo-Z (cheap/very cheap (DIY possible)) or (Hi-Z) FET.

Raymond