Unless there is some national publicly available elevation data (which I doubt), your best bet might be SRTM - 30m resolution and more or less global except at high latitudes.
The USGS website used to have LANDSAT DATA on their website but requires clean up before use. However, you are better off purchasing clean data by contacting USGS directly. They can mail it to you in CD or other formats. Good luck.
I've been using EarthExplorer link which Dr. Nataliia has mentioned in the first answer for some years now for all my research work. The last time I checked for Landsat Satellite data for Republic of Yemen, they had freely available satellite Images for the last 43 years (1972 - present). As Mr. Barrett mentioned you can also find multiple options for Digital Elevation Models (DEM) for free here. Not only Landsat but other multi-spectral satellite data such as MODIS, ASTER etc. Even you can get hyper-spectral data for some part of Yemen for free and not to mention the aerial photographs from the II World War times are also available for free. There are other data access webpages whose links I have attached have other data for Yemen. Mostly all the Satellite Images are freely available. I think the real question should be how can you process so much Gigabytes of free data rather than if you have data at all for Yemen!
You need to create an account and then start downloading the Images. For Landsat, you can pan-sharpen landsat 8 satellite Images to resolutions of 15 metres. As far as topographical data for Yemen is concerned you can access freely the online Map collection from the University of Texas at Austin where they have topographical maps, country maps, detailed maps, historical maps, thematic maps etc. for Yemen! Downloading the data is the easier part; processing, analyzing and interpreting the data is the challenging part.
In today's data rich free internet environment what else can you ask for? If you still need anymore help, ask your student to contact me.
I am also a regular user of landsat data, it's very enriching and hundred percent free, try to log on either www.earthexplorer.uses.gov or http/glovis.usgs.gov
For the topographic map of Yemen, I think you can get that from survey of Yemen or any recognised cartographic institution or government surveys agencies. Then scan and georeference it to the appropriate datum and ellipsoid.