rule = {beta -> 4, alpha -> 2, x -> 4, z -> 2, w -> 0.8}
Select[rule, beta]
Select[rule, beta &]
Select[rule, beta -> _ &]

rule = {beta -> 4, alpha -> 2, x -> 4, z -> 2, w -> 0.8};

FilterRules[rule, beta]
(* {beta -> 4} *)

FilterRules[rule, {beta, alpha}]
(* {beta -> 4, alpha -> 2} *)

KeyTake[rule,{alpha, x}]
(* or *) KeyTake[{alpha,x}][rule]
(* <|alpha->2,x->4|> *)
Normal@KeyTake[rule,{alpha, x}]
(*  {alpha->2,x->4} *)

KeySelect[rule, MatchQ[#,alpha|x]&]
(* or *) KeySelect[MatchQ[#,alpha|x]&][rule]
(* <|alpha->2,x->4|>  *)
Normal@KeySelect[rule, MatchQ[#,alpha|x]&]
(*  {alpha->2,x->4} *)

Select[rule, MemberQ[{beta, alpha}, #[[1]]] &]

Cases[rule, _[beta | alpha, _]]

{beta -> 4, alpha -> 2}

{beta, alpha} /. (# -> Rule[##] & @@@ rule)

Pick[rule, Keys@rule, alpha | beta]

{beta -> 4, alpha -> 2}

Rule[#, (Association@rule)[#]] & /@ {beta, alpha}