From Cache how to Retrieve and Use/Reuse a Bearer Token to authenticate and send data to REST web service?

For the first part of your question, here's a sample method that gets a bearer token from the request:

ClassMethod GetAccessTokenFromRequest(pRequest As %CSP.Request = {%request}) As %String
{
    Set accessToken=""
    Set authorizationHeader=pRequest.GetCgiEnv("HTTP_AUTHORIZATION")
    If $zcvt($piece(authorizationHeader," ",1),"U")="BEARER" {
        If $length(authorizationHeader," ")=2 {
            Set accessToken=$piece(authorizationHeader," ",2)
        }
    }
    return accessToken
}

EDIT:
And here is a full sample of a REST handler that retrieves a bearer token and reuses it to make a request against another REST service.

Class API.DemoBearerToken Extends %CSP.REST
{

Parameter APIHOST = "localhost";

Parameter APIPORT = 52773;

Parameter APIPATH = "/some/other/path";

XData UrlMap [ XMLNamespace = "http://www.intersystems.com/urlmap" ]
{
<Routes>
<Route Url="/example" Method="GET" Call="example"/>
</Routes>
}

ClassMethod example()
{
    set accessToken = ..GetAccessTokenFromRequest(%request)
    set req = ##class(%Net.HttpRequest).%New()
    set req.Https = 1
    set req.SSLConfiguration = "some ssl config"
    set req.Server = ..#APIHOST
    set req.Port = ..#APIPORT
    set req.Authorization = "Bearer "_accessToken
    $$$ThrowOnError(req.Get(..#APIPATH))
    set %response.Status = req.HttpResponse.StatusCode
    set %response.ContentType = req.HttpResponse.ContentType
    if req.HttpResponse.StatusCode <= 400 { //if not an error response
        set jsonData = {}.%FromJSON(req.HttpResponse.Data)
        write jsonData.%ToJSON()
    }
    return $$$OK
}

ClassMethod GetAccessTokenFromRequest(pRequest As %CSP.Request = {%request}) As %String
{
    Set accessToken=""
    Set authorizationHeader=pRequest.GetCgiEnv("HTTP_AUTHORIZATION")
    If $zcvt($piece(authorizationHeader," ",1),"U")="BEARER" {
        If $length(authorizationHeader," ")=2 {
            Set accessToken=$piece(authorizationHeader," ",2)
        }
    }
    return accessToken
}

}