What measures can be taken at home to relieve sinus pain?

Various. Could consider oral decongestants like sudafed, nasal spray decongestants like oxymetazoline, nasal saline sprays or rinses. Each of these treatments have instructions and warnings that need to be read and heeded.
Hot moist towels. Again, anything that increases your circulation will speed up and increase healing. Use hot most towels over the face to increase the circulation. Although this sounds a bit to the contrary, using a spray decongestant such as afrin, neo-synephrine, 4-way, or Vicks for a few days can constrict the swollen vessels in the mucous membranes of the nasal cavity and sinus openings and give relief. Drj.