Lord Shiva himself is the controller of time. So, there is no use of waiting for specific time while being devoted towards Lord Shiva. But still if specific time is choosen then morning after rising from the bed is considered best. Lord Krishna himself also used to chant Sri Rudram daily after rising from bed in the morning:
Vasudeva said, 'I shall recite to thee the good that I have acquired and the fame that I have won through the grace of that high-souled one. Verily, I shall discourse to thee on the topic, after having bowed unto Kapardin. O king, listen to me as I recite to thee that Sata-rudriya which I repeat; with restrained senses, every morning after rising from bed. The great lord of all creatures, viz., the Grandsire Brahman himself, endued with wealth of penances, composed those Mantras, after having observed especial penances for some time. O sire it is Sankara who created all the creatures in the universe, mobile and immobile. There is no being that is higher, O monarch, than Mahadeva. Verily, he is the highest of all beings in the three worlds........ (Mahabharata Anushashana Parva CLX)
For rituals it may differ from people. You can simply chant it being devoted to Lord Shiva. Some also use special rituals using Karanyas-Anganyas, Kavacham and so on. If you want these Karanyas Anganyas and Kavacha procedures and their order of chanting, you can see this video. It isn't necessary that you should stand in river and chant it.
For the benefits of chanting Sri Rudram, Kaivalya Upanishad says:
- He who studies the Shatarudriya, is purified as by the Fires, is purified from the sin of drinking, purified from the sin of killing a Brahmana, from deeds done knowingly or unawares. Through this he has his refuge in Shiva, the Supreme Self. One who belongs to the highest order of life should repeat this always or once (a day).
- By means of this, one attains the Knowledge that destroys the ocean of Samsara or repeated transmigration. Therefore, knowing thus one attains the fruit of Kaivalya or liberation, verily one attains liberation.