You could use javascript for that. There are several possibilities, one way would be to get the element and use scrollIntoView(), see anchor jumping by using javascript.
An easy way to use javascript in shiny, is library(shinyjs).
You could insert the following to tell R to move the element in focus:
runjs('
      document.getElementById("anchor_box").scrollIntoView();
    ')
In order to know, when to do so, you could wrap it within a observeEvent(): 
observeEvent(eventExpr = input$clck, handlerExpr = {...})
Downside would be that refreshing the page would not automatically "scroll up to the top".
Reproducible example:
library(shiny)
library(shinyjs)
ui <- fluidPage(
  a("Go to Results", href = '#anchor_box'),
  actionButton("clck", "Move Programmatically!"),
  plotOutput("plot", height = 1300),  
  div(id = "anchor_box", "HERE"),
  useShinyjs(),
)
server <- function(input, output, session) {
  output$plot <- renderPlot({
    plot(1)
  })
  observeEvent(eventExpr = input$clck, handlerExpr = {
    runjs('
      document.getElementById("anchor_box").scrollIntoView();
    ')
  })
}
shinyApp(ui, server)