Arsenal have been in strong form, unbeaten in the league since November and recently thrashing Manchester City 5-1.
With exits from both domestic cups, Mikel Arteta's side can now shift more focus toward their Premier League campaign.
"There's still a really long way to go," Rice told Sky Sports.
"The manager said in a press conference the other week, no matter what, the only thing we can worry about is ourselves.
"But at the end of the season, we will be there. We can only keep believing in what we believe in, work on what we're working on and have full trust in the manager, players and everyone around us that we can achieve great things.
"It's down to us to get that over the line. Big things are coming. We need to stay positive and keep going."