Your question is actually three questions. The first is simple and factual to answer; the second is based on possibly faulty assumptions; and the third has been asked before here.

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Answering your first question, "What kind of lens is used in this picture of Earth?", from an answer to a different question about the moon missions,

for the Apollo mission images, the camera and lens information is well documented. The still photos taken by the lunar module team were taken exclusively by Hasselblad cameras modified to accept 70 mm film backs, 1:1 image ratio, with 60mm Zeiss Metric lenses.

• Photography Equipment and Techniques, A Survey of NASA Developments, 1972.

From NASA's history page, Astronaut Still Photography During Apollo,

Five more flights landed on the Moon after Apollo 11. On all, the photographic equipment and films were similar to that taken on the first landing. On Apollo 15, the 250mm telescopic lens was added to the Hasselblad lunar surface complement.

See also the wonderful history.nasa.gov resources,

• Apollo Flight Journal
• Apollo Lunar Surface Journal, such as the Apollo 11 Hasselblad Cameras page.

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Regarding the second issue in your question, you said,

I suspect that the lens used doesn't reflect how our eyes would see the Earth from the moon.

I'm not quite sure what you mean by "doesn't reflect how our eyes would see the Earth from the moon". If perhaps you mean apparent size of the Earth, then you need to understand angle and field of view.

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Regarding your third issue/question,

Is there a way by looking or measuring a picture to determine what type of lens is used?

Assuming you're talking inspect only the image elements (and not metadata such as EXIF data embedded in a digital image's file), you need to know a few things:

1. The real-world size of object(s) in the scene.
2. Distances between objects in the scene, or...
3. Distances between the objects in the scene and the camera
4. Whether or not the image has been cropped, enlarged, etc.
5. Information about the size of the film or image sensor used to take the image.

Using this information, you can probably deduce the focal length of the lens used to take the image. The more information you know, and the higher the quality of that information, the better your guess can be to determine the lens's focal length.

The diameter of the earth is roughly 12,740 km. The earth and moon are about 384,400 km apart. Therefore, from the moon's surface, the earth appears to cover 2° of arc in the moon's "sky". In your image (which is 3804 px wide), I figure the earth is about 170 px across. So there are about 22.3 earth radii across the image. So the image covers a view of about 22.3 * 2° = 44.6° across.

Using the pinhole projection formula for angle of view, and knowing that the image frame size of the 70mm film on Apollo missions was 56mm across, I calculate the focal length of the lens to be 61mm. This agrees with the known, nominal 60mm focal length of the lenses used in Apollo missions.

Related: What is the formula for percent of frame filled at a specific distance and focal length?