I am new in the world of Radio Technology and I would build an idea with Bluetooth 4.0 and want to see the frequency for debugging. So I will need a Spectrum Analyzer, but won't buy one of these. Bluetooth 4.0 works with 2,4GHz and that is now my question. Can I build a Spectrum Analyzer with a wireless antenna and maybe setting the interface in passive mode for this kind of action?
Probably not. Most radio equipment of this sort only gives access to the demodulated-and-decoded signal.
The requirements for being an (uncalibrated) spectrum analyzer of this sort are essentially the same as being used for software-defined radio. So if any particular cheap consumer hardware is usable in the way you want, we would have heard of it as possible SDR equipment. And I haven't heard of any repurposed Bluetooth (or WiFi) devices.
What you can do, which is still cheaper than a proper spectrum analyzer, is get a hobbyist SDR device that covers 2.4 GHz, such as the HackRF One or bladeRF.
In any case, the primary thing that you lose by not using a real spectrum analyzer is calibration — being able to determine the true power of an observed signal at any frequency. The lower-cost equipment will still be able to show you the presence or absence of a signal, and its strength relative to other signals at the same frequency.
In addition to what AG6YO said in his answer (No, a network card is a network card and is built to find networks, not "spurious" power of another standard) – Bluetooth as is is a protocol especially hard to capture with a spectrum analyzer.
The reason is very simple: Bluetooth works by jumping through separate, small 1MHz wide channels within a large overall bandwidth (up to 79 MHz IIRC). That hopping happens at up to 1600 hops per second.
Now, there's specific exceptions to that in the Bluetooth standard: Bluetooth LE only uses a single 1MHz bandwidth – but you only mention you're using Bluetooth 4.0, which is the most recent release of Bluetooth, and contains all low- to extremely high rate-modes.
At the same time, classically, spectrum analyzers work by "tuning" in sequentially on small parts of the spectrum, measuring the power that's passing through a narrow filter (the narrower the filter, the higher the spectral resolution) for a short duration of time, moving on to the next frequency and so on.
Hence, for sweeping spectrum analyzers (SAs), the intercept probability of bluetooth is really low. Without a lot of luck and/or experience, you won't be able to configure your SA to even stand a chance of noticing bluetooth, let alone debug it.
So, a spectrum analyzer is clearly not the only device you'd need for bluetooth debugging. As Kevin pointed out, the right choice will be an SDR – something that continously samples what signal is on the air over a large bandwidth, giving it to a computer, so that software there can either generate a visualization of all the spectrum at once (as opposed to SAs, which visualize spectrum in small steps, but with more precision / higher dynamic range), or even try to decipher the Bluetooth communication.
If you can't restrict how much your Bluetooth system will jump (in general, you can't), however, you will need to cover all of the aforementioned 80 MHz at once. Which means you'd need an SDR device with at least that bandwidth – and that's just the beginning of your fun, since actually processing let's say 100MS/s isn't going to be easy on any hardware, and even just getting these samples out of the SDR device into e.g. a PC is a costly feat.
For the 1MHz-only Bluetooth variants, however, the devices mentioned (or basically, most SDRs that cover 2.4GHz) could do the job – a hackRF, the ubertooth, a lot of the more costly USRPs etc.
That still leaves one question open: What is it that you want to debug? Bluetooth is still a complex standard, and though you can capture a few modes and even extract the raw digital packets from the analog signal, my blind guess is that this kind of approach to understanding what happens between your bluetooth devices might be a bit too low-level to actually help you advance, unless you already know well enough what you're looking for.