Late arrival at Narita airport

I am landing at Narita Airport at 9pm. It would be app. 10pm or later by the time I finish the formalities. The last Narita express leaves at 945pm; is it correct? If yes, what are the other cheap options to get to Shinjuku? Is there any metro connection we can take like the JR Sobu line or Keisei limited express.

HYPERDIA.

The last Keisei train running from NRT for (Keisei) Ueno is to depart around 22:40.
: You can make a transfer from Keisei to JR in Nippori, passing through one transfer gate.

The last JR train running from NRT for Tokyo is to depart around 23:00.
: If you take this JR train, you will need to make two transfers, first changing platforms and second at the same platform.

Railroad stations in NRT Terminal 1 and 2 are on the 1st basement.

The last limousine bus running from NRT for Shinjuku is to depart a little past 23:00.
: The one-way fare is 3100 yen per adult.
: You cannot book this limousine bus.
: The bus terminal is actually a bit apart from Shinjuku Station of JR.
http://www.limousinebus.co.jp/en/platform_searches/index/2/212

If your Shinjuku hotel is around either Shinjuku Station or the bus terminal but not near the both, you had better expect about 1000 yen per car for a taxi ride. Usually one taxi car can carry up to 4 persons.

[As of April 2014: The last 4 railroad connections from NRT to Shinjuku Sta.]

Except in Skyliner 56, whose all seats require reservation, you cannot reserve your seats in these trains.
Regarding Keisei rides, you will need to specify your first line, when you buy your ticket, which should be either Narita SKY ACCESS Line (in Case (A) / (B)) or Keisei Main Line (in Case (C)).

(A) {The 4th last railroad connection: 2670 yen per adult}
Narita Airport (22:30) / Airport Terminal 2 (22:33)
: Before the departure, you will need to reserve your seat, paying 1230 yen per adult as a Skyliner charge.
: Here you will pass through the Keisei entrance gate only.
--[Narita SKY ACCESS Line + Keisei Main Line: Skyliner 56]--
(23:09) Nippori --Transfer via the transfer gate for JR lines--
Nippori: Track 11 (around 23:15) --[JR Yamanote Linefs inner loop]--
(around 23:40) Shinjuku.

(B) {The 3rd last railroad connection: 1440 yen per adult}
Narita Airport (22:34) / Airport Terminal 2 (22:37)
: Here you will pass through the Keisei entrance gate only.
--[Narita SKY ACCESS Line + Keisei Main Line: Access Express]--
(around 23:30) Nippori --Transfer via the transfer gate for JR lines--
Nippori: Track 11 (around 23:40) --[JR Yamanote Linefs inner loop]--
(around 00:05) Shinjuku.

(C) {The 2nd last railroad connection: 1230 yen per adult}
Narita Airport (22:39) / Airport Terminal 2 (22:42)
: Here you will pass through both the entrance gate and the intermediate gate.
--[Keisei Main Line: Commuter Express]--
(around 23:55) Nippori --Transfer via the transfer gate for JR lines--
Nippori: Track 11 (around 00:05) --[JR Yamanote Linefs inner loop]--
(around 00:30) Shinjuku.

(D) {The last railroad connection: 1490 yen per adult}
Narita Airport (23:00) / Airport Terminal 2 (23:02)
: I recommend getting aboard in its front half, for a quick transfer in Kinshicho.
--[JR lines: Rapid]--
(00:21) Kinshicho --Transfer to the other platform using escalators--
(00:32) --[JR: Local]--
(00:40) Ochanomizu --Transfer at the same platform--
(00:43) --[JR Chuo Line]--
(01:01) Shinjuku.

[Narita Express]

The last Narita express leaves at 9:45 pm; is it correct?

Yes, the last Narita Express train running for Tokyo is scheduled to depart at 21:44 from Narita Airport Station / at 21:45 from Airport Terminal 2 Station.
Like many other NfEX trains, itfs a two-destination train until reaching Tokyo Station, where it is to be divided into two parts. One of its parts is bound for Ikebukuro and scheduled to arrive to Shinjuku at 23:02.

[Railroad in and around Tokyo]

Is there any metro connection we can take like the JR Sobu line or Keisei limited express.

A line of JR-East / Keisei is not called "metro" or "subway" even when it includes a part under the ground.

The term "Metro" may refer to Tokyo Metro, which is one of the two operators of subway lines in and around Tokyo.
There are nine Tokyo Metro subway lines and four Toei subway lines. "Subway" is a general term for them.
Some subway lines lie partly on or above the ground.
- Tokyo Subway Route Map
http://www.tokyometro.jp/en/subwaymap/pdf/routemap_en.pdf
: You see Shinjuku (M-08, S-01, E-27) in the left half.

Sobu Main Line of JR-East is long and partly has two legs.
The phrase "Sobu Line" may refer to different parts of this line.

JR Rapid trains and Narita Express trains to and from NRT take the same route at least between Tokyo and Narita Airport.
Tokyo: Tracks on the 5th basement --[Sobu Line (leg for faster trains)]-- Kinshicho --[Sobu Line]-- Chiba --[Sobu (Main) Line]-- Sakura --[Narita Line]-- Narita -- [Narita Line (leg for NRT)] -- Narita Airport.

Except in very late / early hours, you can take a direct train between Shinjuku and Tokyo on Chuo Line (pronounced like chew-all-line).
Shinjuku --[Chuo Line]-- Ochanomizu --[Chuo Line]-- Tokyo: Tracks above the ground.
: This route is mainly for rapid trains.
: This route is shorter than routes by Yamanote Line.

Please note that there are also trains on Chuo Line which do not run via Tokyo Station.
Shinjuku --[Chuo Line]-- Ochanomizu --[Sobu Line (leg for Local trains)]-- Kinshicho --[Sobu Line: Local]-- Chiba.
: This route for is for Local trains and is sometimes called Chuo Sobu Line.