How is one to clean his hands? [1]
[Although it is necessary to wash one’s hands with water in order to remove the impurity that resides on the hands after some[2] of the above actions were performed nevertheless,] to be allowed to learn Torah and recite blessings it is not necessary to specifically wash the hands with water. Rather, he may clean them with anything that cleans.[3] This applies even if one has water readily available in front of him, nevertheless he may clean them in other ways.[4] However, prior to prayer one is required to wash his hands specifically with water.[5]
[1] Admur Basra 4:1; 7:1; 92:6; Siddur Seder Netila; M”B 4:61; 227:11
In 613:4 Admur allows washing with water even on Yom Kippur, if one got his hands dirty in the process. This however does not mean to say that using water is an obligation, but rather simply that it is allowed. This also explains why he uses the word “permitted” and not “obligated” regarding this washing. However Tzaruch Iyun from Seder Birchas Hanehnin 13:17 where Admur rules that one who used the bathroom is required to wash his hands prior to saying the blessing over thunder and does not offer the ability to simply clean the hands on another item. The M”B 227:11 writes regarding that Halacha that one can either wash his hands or clean them in another method, and so writes Piskeiy Teshuvos 227:8. Vetzaruch Iyun.
[2] All matters that were listed above in B that were not listed in Halacha 1 to cause impure spirit, do not cause it and one is hence not required to clean his hands specifically with water. With regards to touching mucus or wax of the ear, Admur does not mention them in the list of matters that require washing. [See Halacha 4P] Nevertheless if one touched the inside of his ear or nose it is considered as if he touched the normally covered areas of the body and hence washing to remove the impure spirit is required. [Piskeiy Teshuvos 4:29]
[3] For this reason, according to all opinions, when washing prior to learning or reciting a blessing one does not say a blessing on the washing. [Basra ibid]
[4] 92:6; Peri Megadim brought in M”B 4:61
[5] Basra 4:1
However even prior to prayer, if there is no water available, one may clean his hands with anything that cleans, as the verse does not specify a cleansing with using water. Nevertheless before prayer it’s a mitzvah:obligation to walk up to a mil’s distance [18-24 minutes] to find water.